Curvilinear coordinates can be formulated in tensor calculus , with important applications in physics and engineering , particularly for describing transportation of physical quantities and deformation of matter in fluid mechanics and continuum mechanics .
Vector and tensor algebra in three-dimensional curvilinear coordinates [ edit ] Elementary vector and tensor algebra in curvilinear coordinates is used in some of the older scientific literature in mechanics and physics and can be indispensable to understanding work from the early and mid 1900s, for example the text by Green and Zerna.[ 1] [ 2] [ 3] [ 4] [ 1] [ 5] [ 6]
Consider two coordinate systems with coordinate variables
(
Z
1
,
Z
2
,
Z
3
)
{\displaystyle (Z^{1},Z^{2},Z^{3})}
(
Z
1
´
,
Z
2
´
,
Z
3
´
)
{\displaystyle (Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})}
Z
i
{\displaystyle Z^{i}}
Z
i
´
{\displaystyle Z^{\acute {i}}}
i
{\displaystyle i}
Z
i
{\displaystyle Z^{i}}
Z
i
´
{\displaystyle Z^{\acute {i}}}
Z
i
{\displaystyle Z^{i}}
Z
i
´
{\displaystyle Z^{\acute {i}}}
Z
i
=
f
i
(
Z
1
´
,
Z
2
´
,
Z
3
´
)
{\displaystyle Z^{i}=f^{i}(Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})}
i
=
1
,
2
,
3
{\displaystyle i=1,2,3}
which can be written as
Z
i
=
Z
i
(
Z
1
´
,
Z
2
´
,
Z
3
´
)
=
Z
i
(
Z
i
´
)
{\displaystyle Z^{i}=Z^{i}(Z^{\acute {1}},Z^{\acute {2}},Z^{\acute {3}})=Z^{i}(Z^{\acute {i}})}
i
´
,
i
=
1
,
2
,
3
{\displaystyle {\acute {i}},i=1,2,3}
These three equations together are also called a coordinate transformation from
Z
i
´
{\displaystyle Z^{\acute {i}}}
Z
i
{\displaystyle Z^{i}}
T
{\displaystyle T}
Z
i
´
{\displaystyle Z^{\acute {i}}}
Z
i
{\displaystyle Z^{i}}
Z
=
T
(
z
´
)
{\displaystyle Z=T({\acute {z}})}
Similarly we can represent
Z
i
´
{\displaystyle Z^{\acute {i}}}
Z
i
{\displaystyle Z^{i}}
Z
i
´
=
g
i
´
(
Z
1
,
Z
2
,
Z
3
)
{\displaystyle Z^{\acute {i}}=g^{\acute {i}}(Z^{1},Z^{2},Z^{3})}
i
´
=
1
,
2
,
3
{\displaystyle {\acute {i}}=1,2,3}
and we can write the free equations more compactly as
Z
i
´
=
Z
i
´
(
Z
1
,
Z
2
,
Z
3
)
=
Z
i
´
(
Z
i
)
{\displaystyle Z^{\acute {i}}=Z^{\acute {i}}(Z^{1},Z^{2},Z^{3})=Z^{\acute {i}}(Z^{i})}
i
´
,
i
=
1
,
2
,
3
{\displaystyle {\acute {i}},i=1,2,3}
These three equations together are also called a coordinate transformation from
Z
i
{\displaystyle Z^{i}}
Z
i
´
{\displaystyle Z^{\acute {i}}}
S
{\displaystyle S}
Z
i
{\displaystyle Z^{i}}
Z
i
´
{\displaystyle Z^{\acute {i}}}
z
´
=
S
(
z
)
{\displaystyle {\acute {z}}=S(z)}
If the transformation
T
{\displaystyle T}
Z
i
{\displaystyle Z^{i}}
admissible coordinates for
Z
i
´
{\displaystyle Z^{\acute {i}}}
T
{\displaystyle T}
Z
i
{\displaystyle Z^{i}}
affine coordinate system , otherwise
Z
i
{\displaystyle Z^{i}}
curvilinear coordinate system .
As we now see that the Coordinates
Z
i
{\displaystyle Z^{i}}
Z
i
´
{\displaystyle Z^{\acute {i}}}
Z
i
{\displaystyle Z^{i}}
Z
i
´
{\displaystyle Z^{\acute {i}}}
Consider
∂
Z
i
∂
Z
i
´
=
d
e
f
J
i
´
i
{\displaystyle {\frac {\partial {Z^{i}}}{\partial {Z^{\acute {i}}}}}\;{\overset {\underset {\mathrm {def} }{}}{=}}\;J_{\acute {i}}^{i}}
i
´
,
i
=
1
,
2
,
3
{\displaystyle {\acute {i}},i=1,2,3}
J
{\displaystyle J}
J
i
´
i
{\displaystyle J_{\acute {i}}^{i}}
i
{\displaystyle i}
i
´
{\displaystyle {\acute {i}}}
J
=
(
J
1
´
1
J
2
´
1
J
3
´
1
J
1
´
2
J
2
´
2
J
3
´
2
J
1
´
3
J
2
´
3
J
3
´
3
)
=
(
∂
Z
1
∂
Z
1
´
∂
Z
1
∂
Z
2
´
∂
Z
1
∂
Z
3
´
∂
Z
2
∂
Z
1
´
∂
Z
2
∂
Z
2
´
∂
Z
2
∂
Z
3
´
∂
Z
3
∂
Z
1
´
∂
Z
3
∂
Z
2
´
∂
Z
3
∂
Z
3
´
)
{\displaystyle J={\begin{pmatrix}J_{\acute {1}}^{1}&J_{\acute {2}}^{1}&J_{\acute {3}}^{1}\\J_{\acute {1}}^{2}&J_{\acute {2}}^{2}&J_{\acute {3}}^{2}\\J_{\acute {1}}^{3}&J_{\acute {2}}^{3}&J_{\acute {3}}^{3}\end{pmatrix}}={\begin{pmatrix}{\partial {Z^{1}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{1}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{1}} \over \partial {Z^{\acute {3}}}}\\{\partial {Z^{2}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{2}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{2}} \over \partial {Z^{\acute {3}}}}\\{\partial {Z^{3}} \over \partial {Z^{\acute {1}}}}&{\partial {Z^{3}} \over \partial {Z^{\acute {2}}}}&{\partial {Z^{3}} \over \partial {Z^{\acute {3}}}}\end{pmatrix}}}
The resultant matrix is called the Jacobian matrix.
Vectors in curvilinear coordinates [ edit ] Let (b 1 , b 2 , b 3 ) be an arbitrary basis for three-dimensional Euclidean space. In general, the basis vectors are neither unit vectors nor mutually orthogonal . However, they are required to be linearly independent. Then a vector v can be expressed as[ 4] : 27
v
=
v
k
b
k
{\displaystyle \mathbf {v} =v^{k}\,\mathbf {b} _{k}}
vk are the contravariant components of the vector v .
The reciprocal basis (b 1 , b 2 , b 3 ) is defined by the relation [ 4] : 28–29
b
i
⋅
b
j
=
δ
j
i
{\displaystyle \mathbf {b} ^{i}\cdot \mathbf {b} _{j}=\delta _{j}^{i}}
δi j is the Kronecker delta .
The vector v can also be expressed in terms of the reciprocal basis:
v
=
v
k
b
k
{\displaystyle \mathbf {v} =v_{k}~\mathbf {b} ^{k}}
vk are the covariant components of the vector
v
{\displaystyle \mathbf {v} }
Second-order tensors in curvilinear coordinates [ edit ] A second-order tensor can be expressed as
S
=
S
i
j
b
i
⊗
b
j
=
S
j
i
b
i
⊗
b
j
=
S
i
j
b
i
⊗
b
j
=
S
i
j
b
i
⊗
b
j
{\displaystyle {\boldsymbol {S}}=S^{ij}~\mathbf {b} _{i}\otimes \mathbf {b} _{j}=S_{~j}^{i}~\mathbf {b} _{i}\otimes \mathbf {b} ^{j}=S_{i}^{~j}~\mathbf {b} ^{i}\otimes \mathbf {b} _{j}=S_{ij}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}}
Sij are called the contravariant components, Si j the mixed right-covariant components, Si j the mixed left-covariant components, and Sij the covariant components of the second-order tensor.
Metric tensor and relations between components [ edit ] The quantities gij , gij are defined as[ 4] : 39
g
i
j
=
b
i
⋅
b
j
=
g
j
i
;
g
i
j
=
b
i
⋅
b
j
=
g
j
i
{\displaystyle g_{ij}=\mathbf {b} _{i}\cdot \mathbf {b} _{j}=g_{ji}~;~~g^{ij}=\mathbf {b} ^{i}\cdot \mathbf {b} ^{j}=g^{ji}}
v
i
=
g
i
k
v
k
;
v
i
=
g
i
k
v
k
;
b
i
=
g
i
j
b
j
;
b
i
=
g
i
j
b
j
{\displaystyle v^{i}=g^{ik}~v_{k}~;~~v_{i}=g_{ik}~v^{k}~;~~\mathbf {b} ^{i}=g^{ij}~\mathbf {b} _{j}~;~~\mathbf {b} _{i}=g_{ij}~\mathbf {b} ^{j}}
The components of a vector are related by[ 4] : 30–32
v
⋅
b
i
=
v
k
b
k
⋅
b
i
=
v
k
δ
k
i
=
v
i
{\displaystyle \mathbf {v} \cdot \mathbf {b} ^{i}=v^{k}~\mathbf {b} _{k}\cdot \mathbf {b} ^{i}=v^{k}~\delta _{k}^{i}=v^{i}}
v
⋅
b
i
=
v
k
b
k
⋅
b
i
=
v
k
δ
i
k
=
v
i
{\displaystyle \mathbf {v} \cdot \mathbf {b} _{i}=v_{k}~\mathbf {b} ^{k}\cdot \mathbf {b} _{i}=v_{k}~\delta _{i}^{k}=v_{i}}
v
⋅
b
i
=
v
k
b
k
⋅
b
i
=
g
k
i
v
k
{\displaystyle \mathbf {v} \cdot \mathbf {b} _{i}=v^{k}~\mathbf {b} _{k}\cdot \mathbf {b} _{i}=g_{ki}~v^{k}}
v
⋅
b
i
=
v
k
b
k
⋅
b
i
=
g
k
i
v
k
{\displaystyle \mathbf {v} \cdot \mathbf {b} ^{i}=v_{k}~\mathbf {b} ^{k}\cdot \mathbf {b} ^{i}=g^{ki}~v_{k}}
The components of the second-order tensor are related by
S
i
j
=
g
i
k
S
k
j
=
g
j
k
S
k
i
=
g
i
k
g
j
l
S
k
l
{\displaystyle S^{ij}=g^{ik}~S_{k}^{~j}=g^{jk}~S_{~k}^{i}=g^{ik}~g^{jl}~S_{kl}}
The alternating tensor [ edit ] In an orthonormal right-handed basis, the third-order alternating tensor is defined as
E
=
ε
i
j
k
e
i
⊗
e
j
⊗
e
k
{\displaystyle {\boldsymbol {\mathcal {E}}}=\varepsilon _{ijk}~\mathbf {e} ^{i}\otimes \mathbf {e} ^{j}\otimes \mathbf {e} ^{k}}
E
=
E
i
j
k
b
i
⊗
b
j
⊗
b
k
=
E
i
j
k
b
i
⊗
b
j
⊗
b
k
{\displaystyle {\boldsymbol {\mathcal {E}}}={\mathcal {E}}_{ijk}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}\otimes \mathbf {b} ^{k}={\mathcal {E}}^{ijk}~\mathbf {b} _{i}\otimes \mathbf {b} _{j}\otimes \mathbf {b} _{k}}
E
i
j
k
=
[
b
i
,
b
j
,
b
k
]
=
(
b
i
×
b
j
)
⋅
b
k
;
E
i
j
k
=
[
b
i
,
b
j
,
b
k
]
{\displaystyle {\mathcal {E}}_{ijk}=\left[\mathbf {b} _{i},\mathbf {b} _{j},\mathbf {b} _{k}\right]=(\mathbf {b} _{i}\times \mathbf {b} _{j})\cdot \mathbf {b} _{k}~;~~{\mathcal {E}}^{ijk}=\left[\mathbf {b} ^{i},\mathbf {b} ^{j},\mathbf {b} ^{k}\right]}
b
i
×
b
j
=
J
ε
i
j
p
b
p
=
g
ε
i
j
p
b
p
{\displaystyle \mathbf {b} _{i}\times \mathbf {b} _{j}=J~\varepsilon _{ijp}~\mathbf {b} ^{p}={\sqrt {g}}~\varepsilon _{ijp}~\mathbf {b} ^{p}}
E
i
j
k
=
J
ε
i
j
k
=
g
ε
i
j
k
{\displaystyle {\mathcal {E}}_{ijk}=J~\varepsilon _{ijk}={\sqrt {g}}~\varepsilon _{ijk}}
E
i
j
k
=
1
J
ε
i
j
k
=
1
g
ε
i
j
k
{\displaystyle {\mathcal {E}}^{ijk}={\cfrac {1}{J}}~\varepsilon ^{ijk}={\cfrac {1}{\sqrt {g}}}~\varepsilon ^{ijk}}
The identity map I defined by
I
⋅
v
=
v
{\displaystyle \mathbf {I} \cdot \mathbf {v} =\mathbf {v} }
[ 4] : 39
I
=
g
i
j
b
i
⊗
b
j
=
g
i
j
b
i
⊗
b
j
=
b
i
⊗
b
i
=
b
i
⊗
b
i
{\displaystyle \mathbf {I} =g^{ij}\mathbf {b} _{i}\otimes \mathbf {b} _{j}=g_{ij}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=\mathbf {b} _{i}\otimes \mathbf {b} ^{i}=\mathbf {b} ^{i}\otimes \mathbf {b} _{i}}
[ edit ] The scalar product of two vectors in curvilinear coordinates is[ 4] : 32
u
⋅
v
=
u
i
v
i
=
u
i
v
i
=
g
i
j
u
i
v
j
=
g
i
j
u
i
v
j
{\displaystyle \mathbf {u} \cdot \mathbf {v} =u^{i}v_{i}=u_{i}v^{i}=g_{ij}u^{i}v^{j}=g^{ij}u_{i}v_{j}}
[ edit ] The cross product of two vectors is given by:[ 4] : 32–34
u
×
v
=
ε
i
j
k
u
j
v
k
e
i
{\displaystyle \mathbf {u} \times \mathbf {v} =\varepsilon _{ijk}u_{j}v_{k}\mathbf {e} _{i}}
where εijk permutation symbol and e i
u
×
v
=
[
(
b
m
×
b
n
)
⋅
b
s
]
u
m
v
n
b
s
=
E
s
m
n
u
m
v
n
b
s
{\displaystyle \mathbf {u} \times \mathbf {v} =[(\mathbf {b} _{m}\times \mathbf {b} _{n})\cdot \mathbf {b} _{s}]u^{m}v^{n}\mathbf {b} ^{s}={\mathcal {E}}_{smn}u^{m}v^{n}\mathbf {b} ^{s}}
where
E
i
j
k
{\displaystyle {\mathcal {E}}_{ijk}}
third-order alternating tensor . The cross product of two vectors is given by:
u
×
v
=
ε
i
j
k
u
^
j
v
^
k
e
i
{\displaystyle \mathbf {u} \times \mathbf {v} =\varepsilon _{ijk}{\hat {u}}_{j}{\hat {v}}_{k}\mathbf {e} _{i}}
where εijk permutation symbol and
e
i
{\displaystyle \mathbf {e} _{i}}
e
p
×
e
q
=
ε
i
p
q
e
i
{\displaystyle \mathbf {e} _{p}\times \mathbf {e} _{q}=\varepsilon _{ipq}\mathbf {e} _{i}}
and
b
m
×
b
n
=
∂
x
∂
q
m
×
∂
x
∂
q
n
=
∂
(
x
p
e
p
)
∂
q
m
×
∂
(
x
q
e
q
)
∂
q
n
=
∂
x
p
∂
q
m
∂
x
q
∂
q
n
e
p
×
e
q
=
ε
i
p
q
∂
x
p
∂
q
m
∂
x
q
∂
q
n
e
i
.
{\displaystyle \mathbf {b} _{m}\times \mathbf {b} _{n}={\frac {\partial \mathbf {x} }{\partial q^{m}}}\times {\frac {\partial \mathbf {x} }{\partial q^{n}}}={\frac {\partial (x_{p}\mathbf {e} _{p})}{\partial q^{m}}}\times {\frac {\partial (x_{q}\mathbf {e} _{q})}{\partial q^{n}}}={\frac {\partial x_{p}}{\partial q^{m}}}{\frac {\partial x_{q}}{\partial q^{n}}}\mathbf {e} _{p}\times \mathbf {e} _{q}=\varepsilon _{ipq}{\frac {\partial x_{p}}{\partial q^{m}}}{\frac {\partial x_{q}}{\partial q^{n}}}\mathbf {e} _{i}.}
Hence,
(
b
m
×
b
n
)
⋅
b
s
=
ε
i
p
q
∂
x
p
∂
q
m
∂
x
q
∂
q
n
∂
x
i
∂
q
s
{\displaystyle (\mathbf {b} _{m}\times \mathbf {b} _{n})\cdot \mathbf {b} _{s}=\varepsilon _{ipq}{\frac {\partial x_{p}}{\partial q^{m}}}{\frac {\partial x_{q}}{\partial q^{n}}}{\frac {\partial x_{i}}{\partial q^{s}}}}
Returning to the vector product and using the relations:
u
^
j
=
∂
x
j
∂
q
m
u
m
,
v
^
k
=
∂
x
k
∂
q
n
v
n
,
e
i
=
∂
x
i
∂
q
s
b
s
,
{\displaystyle {\hat {u}}_{j}={\frac {\partial x_{j}}{\partial q^{m}}}u^{m},\quad {\hat {v}}_{k}={\frac {\partial x_{k}}{\partial q^{n}}}v^{n},\quad \mathbf {e} _{i}={\frac {\partial x_{i}}{\partial q^{s}}}\mathbf {b} ^{s},}
gives us:
u
×
v
=
ε
i
j
k
u
^
j
v
^
k
e
i
=
ε
i
j
k
∂
x
j
∂
q
m
∂
x
k
∂
q
n
∂
x
i
∂
q
s
u
m
v
n
b
s
=
[
(
b
m
×
b
n
)
⋅
b
s
]
u
m
v
n
b
s
=
E
s
m
n
u
m
v
n
b
s
{\displaystyle \mathbf {u} \times \mathbf {v} =\varepsilon _{ijk}{\hat {u}}_{j}{\hat {v}}_{k}\mathbf {e} _{i}=\varepsilon _{ijk}{\frac {\partial x_{j}}{\partial q^{m}}}{\frac {\partial x_{k}}{\partial q^{n}}}{\frac {\partial x_{i}}{\partial q^{s}}}u^{m}v^{n}\mathbf {b} ^{s}=[(\mathbf {b} _{m}\times \mathbf {b} _{n})\cdot \mathbf {b} _{s}]u^{m}v^{n}\mathbf {b} ^{s}={\mathcal {E}}_{smn}u^{m}v^{n}\mathbf {b} ^{s}}
The identity map
I
{\displaystyle {\mathsf {I}}}
I
⋅
v
=
v
{\displaystyle {\mathsf {I}}\cdot \mathbf {v} =\mathbf {v} }
[ 4] : 39
I
=
g
i
j
b
i
⊗
b
j
=
g
i
j
b
i
⊗
b
j
=
b
i
⊗
b
i
=
b
i
⊗
b
i
{\displaystyle {\mathsf {I}}=g^{ij}\mathbf {b} _{i}\otimes \mathbf {b} _{j}=g_{ij}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=\mathbf {b} _{i}\otimes \mathbf {b} ^{i}=\mathbf {b} ^{i}\otimes \mathbf {b} _{i}}
Action of a second-order tensor on a vector [ edit ] The action
v
=
S
u
{\displaystyle \mathbf {v} ={\boldsymbol {S}}\mathbf {u} }
v
i
b
i
=
S
i
j
u
j
b
i
=
S
j
i
u
j
b
i
;
v
i
b
i
=
S
i
j
u
i
b
i
=
S
i
j
u
j
b
i
{\displaystyle v^{i}\mathbf {b} _{i}=S^{ij}u_{j}\mathbf {b} _{i}=S_{j}^{i}u^{j}\mathbf {b} _{i};\qquad v_{i}\mathbf {b} ^{i}=S_{ij}u^{i}\mathbf {b} ^{i}=S_{i}^{j}u_{j}\mathbf {b} ^{i}}
Inner product of two second-order tensors [ edit ] The inner product of two second-order tensors
U
=
S
⋅
T
{\displaystyle {\boldsymbol {U}}={\boldsymbol {S}}\cdot {\boldsymbol {T}}}
U
i
j
b
i
⊗
b
j
=
S
i
k
T
.
j
k
b
i
⊗
b
j
=
S
i
.
k
T
k
j
b
i
⊗
b
j
{\displaystyle U_{ij}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=S_{ik}T_{.j}^{k}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}=S_{i}^{.k}T_{kj}\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}}
Alternatively,
U
=
S
i
j
T
.
n
m
g
j
m
b
i
⊗
b
n
=
S
.
m
i
T
.
n
m
b
i
⊗
b
n
=
S
i
j
T
j
n
b
i
⊗
b
n
{\displaystyle {\boldsymbol {U}}=S^{ij}T_{.n}^{m}g_{jm}\mathbf {b} _{i}\otimes \mathbf {b} ^{n}=S_{.m}^{i}T_{.n}^{m}\mathbf {b} _{i}\otimes \mathbf {b} ^{n}=S^{ij}T_{jn}\mathbf {b} _{i}\otimes \mathbf {b} ^{n}}
Determinant of a second-order tensor [ edit ] If
S
{\displaystyle {\boldsymbol {S}}}
determinant is defined by the relation
[
S
u
,
S
v
,
S
w
]
=
det
S
[
u
,
v
,
w
]
{\displaystyle \left[{\boldsymbol {S}}\mathbf {u} ,{\boldsymbol {S}}\mathbf {v} ,{\boldsymbol {S}}\mathbf {w} \right]=\det {\boldsymbol {S}}\left[\mathbf {u} ,\mathbf {v} ,\mathbf {w} \right]}
where
u
,
v
,
w
{\displaystyle \mathbf {u} ,\mathbf {v} ,\mathbf {w} }
[
u
,
v
,
w
]
:=
u
⋅
(
v
×
w
)
.
{\displaystyle \left[\mathbf {u} ,\mathbf {v} ,\mathbf {w} \right]:=\mathbf {u} \cdot (\mathbf {v} \times \mathbf {w} ).}
Relations between curvilinear and Cartesian basis vectors [ edit ] Let (e 1 , e 2 , e 3 ) be the usual Cartesian basis vectors for the Euclidean space of interest and let
b
i
=
F
e
i
{\displaystyle \mathbf {b} _{i}={\boldsymbol {F}}\mathbf {e} _{i}}
F i e i b i
b
i
⊗
e
i
=
(
F
e
i
)
⊗
e
i
=
F
(
e
i
⊗
e
i
)
=
F
.
{\displaystyle \mathbf {b} _{i}\otimes \mathbf {e} _{i}=({\boldsymbol {F}}\mathbf {e} _{i})\otimes \mathbf {e} _{i}={\boldsymbol {F}}(\mathbf {e} _{i}\otimes \mathbf {e} _{i})={\boldsymbol {F}}~.}
b
i
=
F
−
T
e
i
;
g
i
j
=
[
F
−
1
F
−
T
]
i
j
;
g
i
j
=
[
g
i
j
]
−
1
=
[
F
T
F
]
i
j
{\displaystyle \mathbf {b} ^{i}={\boldsymbol {F}}^{-{\rm {T}}}\mathbf {e} ^{i}~;~~g^{ij}=[{\boldsymbol {F}}^{-{\rm {1}}}{\boldsymbol {F}}^{-{\rm {T}}}]_{ij}~;~~g_{ij}=[g^{ij}]^{-1}=[{\boldsymbol {F}}^{\rm {T}}{\boldsymbol {F}}]_{ij}}
J
:=
det
F
{\displaystyle J:=\det {\boldsymbol {F}}}
[
b
1
,
b
2
,
b
3
]
=
det
F
[
e
1
,
e
2
,
e
3
]
.
{\displaystyle \left[\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right]=\det {\boldsymbol {F}}\left[\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\right]~.}
[
e
1
,
e
2
,
e
3
]
=
1
{\displaystyle \left[\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\right]=1}
J
=
det
F
=
[
b
1
,
b
2
,
b
3
]
=
b
1
⋅
(
b
2
×
b
3
)
{\displaystyle J=\det {\boldsymbol {F}}=\left[\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right]=\mathbf {b} _{1}\cdot (\mathbf {b} _{2}\times \mathbf {b} _{3})}
First, consider
g
:=
det
[
g
i
j
]
{\displaystyle g:=\det[g_{ij}]}
g
=
det
[
F
T
]
⋅
det
[
F
]
=
J
⋅
J
=
J
2
{\displaystyle g=\det[{\boldsymbol {F}}^{\rm {T}}]\cdot \det[{\boldsymbol {F}}]=J\cdot J=J^{2}}
det
[
g
i
j
]
=
1
J
2
{\displaystyle \det[g^{ij}]={\cfrac {1}{J^{2}}}}
[
g
i
j
]
=
[
g
i
j
]
−
1
{\displaystyle [g^{ij}]=[g_{ij}]^{-1}}
∂
g
∂
g
i
j
=
2
J
∂
J
∂
g
i
j
=
g
g
i
j
{\displaystyle {\cfrac {\partial g}{\partial g_{ij}}}=2~J~{\cfrac {\partial J}{\partial g_{ij}}}=g~g^{ij}}
Another interesting relation is derived below. Recall that
b
i
⋅
b
j
=
δ
j
i
⇒
b
1
⋅
b
1
=
1
,
b
1
⋅
b
2
=
b
1
⋅
b
3
=
0
⇒
b
1
=
A
(
b
2
×
b
3
)
{\displaystyle \mathbf {b} ^{i}\cdot \mathbf {b} _{j}=\delta _{j}^{i}\quad \Rightarrow \quad \mathbf {b} ^{1}\cdot \mathbf {b} _{1}=1,~\mathbf {b} ^{1}\cdot \mathbf {b} _{2}=\mathbf {b} ^{1}\cdot \mathbf {b} _{3}=0\quad \Rightarrow \quad \mathbf {b} ^{1}=A~(\mathbf {b} _{2}\times \mathbf {b} _{3})}
A is a, yet undetermined, constant. Then
b
1
⋅
b
1
=
A
b
1
⋅
(
b
2
×
b
3
)
=
A
J
=
1
⇒
A
=
1
J
{\displaystyle \mathbf {b} ^{1}\cdot \mathbf {b} _{1}=A~\mathbf {b} _{1}\cdot (\mathbf {b} _{2}\times \mathbf {b} _{3})=AJ=1\quad \Rightarrow \quad A={\cfrac {1}{J}}}
b
1
=
1
J
(
b
2
×
b
3
)
;
b
2
=
1
J
(
b
3
×
b
1
)
;
b
3
=
1
J
(
b
1
×
b
2
)
{\displaystyle \mathbf {b} ^{1}={\cfrac {1}{J}}(\mathbf {b} _{2}\times \mathbf {b} _{3})~;~~\mathbf {b} ^{2}={\cfrac {1}{J}}(\mathbf {b} _{3}\times \mathbf {b} _{1})~;~~\mathbf {b} ^{3}={\cfrac {1}{J}}(\mathbf {b} _{1}\times \mathbf {b} _{2})}
ε
i
j
k
b
k
=
1
J
(
b
i
×
b
j
)
=
1
g
(
b
i
×
b
j
)
{\displaystyle \varepsilon _{ijk}~\mathbf {b} ^{k}={\cfrac {1}{J}}(\mathbf {b} _{i}\times \mathbf {b} _{j})={\cfrac {1}{\sqrt {g}}}(\mathbf {b} _{i}\times \mathbf {b} _{j})}
ε
i
j
k
{\displaystyle \varepsilon _{ijk}}
permutation symbol .
We have not identified an explicit expression for the transformation tensor F
b
i
=
∂
x
∂
q
i
=
∂
x
∂
x
j
∂
x
j
∂
q
i
=
e
j
∂
x
j
∂
q
i
{\displaystyle \mathbf {b} _{i}={\cfrac {\partial \mathbf {x} }{\partial q^{i}}}={\cfrac {\partial \mathbf {x} }{\partial x_{j}}}~{\cfrac {\partial x_{j}}{\partial q^{i}}}=\mathbf {e} _{j}~{\cfrac {\partial x_{j}}{\partial q^{i}}}}
e
i
=
b
j
∂
q
j
∂
x
i
{\displaystyle \mathbf {e} _{i}=\mathbf {b} _{j}~{\cfrac {\partial q^{j}}{\partial x_{i}}}}
e
k
⋅
b
i
=
∂
x
k
∂
q
i
⇒
∂
x
k
∂
q
i
b
i
=
e
k
⋅
(
b
i
⊗
b
i
)
=
e
k
{\displaystyle \mathbf {e} ^{k}\cdot \mathbf {b} _{i}={\frac {\partial x_{k}}{\partial q^{i}}}\quad \Rightarrow \quad {\frac {\partial x_{k}}{\partial q^{i}}}~\mathbf {b} ^{i}=\mathbf {e} ^{k}\cdot (\mathbf {b} _{i}\otimes \mathbf {b} ^{i})=\mathbf {e} ^{k}}
b
k
=
∂
q
k
∂
x
i
e
i
{\displaystyle \mathbf {b} ^{k}={\frac {\partial q^{k}}{\partial x_{i}}}~\mathbf {e} ^{i}}
Vector and tensor calculus in three-dimensional curvilinear coordinates [ edit ] Simmonds,[ 4] tensor analysis , quotes Albert Einstein saying[ 7]
The magic of this theory will hardly fail to impose itself on anybody who has truly understood it; it represents a genuine triumph of the method of absolute differential calculus, founded by Gauss, Riemann, Ricci, and Levi-Civita.
Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity ,[ 8] mechanics of curved shells ,[ 6] invariance properties of Maxwell's equations which has been of interest in metamaterials [ 9] [ 10]
Some useful relations in the calculus of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden,[ 2] [ 4] [ 1] [ 5] [ 6]
Let the position of a point in space be characterized by three coordinate variables
(
q
1
,
q
2
,
q
3
)
{\displaystyle (q^{1},q^{2},q^{3})}
The coordinate curve q 1 represents a curve on which q 2 , q 3 are constant. Let x be the position vector of the point relative to some origin. Then, assuming that such a mapping and its inverse exist and are continuous, we can write [ 2] : 55
x
=
φ
(
q
1
,
q
2
,
q
3
)
;
q
i
=
ψ
i
(
x
)
=
[
φ
−
1
(
x
)
]
i
{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})~;~~q^{i}=\psi ^{i}(\mathbf {x} )=[{\boldsymbol {\varphi }}^{-1}(\mathbf {x} )]^{i}}
i x ) are called the curvilinear coordinate functions of the curvilinear coordinate system ψ (x ) = φ −1 (x ).
The qi coordinate curves are defined by the one-parameter family of functions given by
x
i
(
α
)
=
φ
(
α
,
q
j
,
q
k
)
,
i
≠
j
≠
k
{\displaystyle \mathbf {x} _{i}(\alpha )={\boldsymbol {\varphi }}(\alpha ,q^{j},q^{k})~,~~i\neq j\neq k}
qj , qk fixed.
Tangent vector to coordinate curves [ edit ] The tangent vector to the curve x i x i qi at the point x ) is
d
x
i
d
α
≡
∂
x
∂
q
i
{\displaystyle {\cfrac {\rm {{d}\mathbf {x} _{i}}}{\rm {{d}\alpha }}}\equiv {\cfrac {\partial \mathbf {x} }{\partial q^{i}}}}
Let f (x ) be a scalar field in space. Then
f
(
x
)
=
f
[
φ
(
q
1
,
q
2
,
q
3
)
]
=
f
φ
(
q
1
,
q
2
,
q
3
)
{\displaystyle f(\mathbf {x} )=f[{\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})]=f_{\varphi }(q^{1},q^{2},q^{3})}
f is defined by
[
∇
f
(
x
)
]
⋅
c
=
d
d
α
f
(
x
+
α
c
)
|
α
=
0
{\displaystyle [{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\rm {d}}{\rm {{d}\alpha }}}f(\mathbf {x} +\alpha \mathbf {c} ){\biggr |}_{\alpha =0}}
c is an arbitrary constant vector. If we define the components ci of c are such that
q
i
+
α
c
i
=
ψ
i
(
x
+
α
c
)
{\displaystyle q^{i}+\alpha ~c^{i}=\psi ^{i}(\mathbf {x} +\alpha ~\mathbf {c} )}
[
∇
f
(
x
)
]
⋅
c
=
d
d
α
f
φ
(
q
1
+
α
c
1
,
q
2
+
α
c
2
,
q
3
+
α
c
3
)
|
α
=
0
=
∂
f
φ
∂
q
i
c
i
=
∂
f
∂
q
i
c
i
{\displaystyle [{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\rm {d}}{\rm {{d}\alpha }}}f_{\varphi }(q^{1}+\alpha ~c^{1},q^{2}+\alpha ~c^{2},q^{3}+\alpha ~c^{3}){\biggr |}_{\alpha =0}={\cfrac {\partial f_{\varphi }}{\partial q^{i}}}~c^{i}={\cfrac {\partial f}{\partial q^{i}}}~c^{i}}
If we set
f
(
x
)
=
ψ
i
(
x
)
{\displaystyle f(\mathbf {x} )=\psi ^{i}(\mathbf {x} )}
q
i
=
ψ
i
(
x
)
{\displaystyle q^{i}=\psi ^{i}(\mathbf {x} )}
[
∇
ψ
i
(
x
)
]
⋅
c
=
∂
ψ
i
∂
q
j
c
j
=
c
i
{\displaystyle [{\boldsymbol {\nabla }}\psi ^{i}(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial \psi ^{i}}{\partial q^{j}}}~c^{j}=c^{i}}
c .
If b i b i
[
∇
f
(
x
)
]
⋅
c
=
∂
f
∂
q
i
c
i
=
(
∂
f
∂
q
i
b
i
)
(
c
i
b
i
)
⇒
∇
f
(
x
)
=
∂
f
∂
q
i
b
i
{\displaystyle [{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial f}{\partial q^{i}}}~c^{i}=\left({\cfrac {\partial f}{\partial q^{i}}}~\mathbf {b} ^{i}\right)\left(c^{i}~\mathbf {b} _{i}\right)\quad \Rightarrow \quad {\boldsymbol {\nabla }}f(\mathbf {x} )={\cfrac {\partial f}{\partial q^{i}}}~\mathbf {b} ^{i}}
A similar process can be used to arrive at the gradient of a vector field f (x ). The gradient is given by
[
∇
f
(
x
)
]
⋅
c
=
∂
f
∂
q
i
c
i
{\displaystyle [{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial \mathbf {f} }{\partial q^{i}}}~c^{i}}
r (x ) = x , then we can show that
c
=
∂
x
∂
q
i
c
i
=
b
i
(
x
)
c
i
;
b
i
(
x
)
:=
∂
x
∂
q
i
{\displaystyle \mathbf {c} ={\cfrac {\partial \mathbf {x} }{\partial q^{i}}}~c^{i}=\mathbf {b} _{i}(\mathbf {x} )~c^{i}~;~~\mathbf {b} _{i}(\mathbf {x} ):={\cfrac {\partial \mathbf {x} }{\partial q^{i}}}}
b i qi coordinate curve and forms a natural basis at each point on the curve. This basis, as discussed at the beginning of this article, is also called the covariant curvilinear basis. We can also define a reciprocal basis , or contravariant curvilinear basis, b i x .
Since c is arbitrary, we can write
∇
f
(
x
)
=
∂
f
∂
q
i
⊗
b
i
{\displaystyle {\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} )={\cfrac {\partial \mathbf {f} }{\partial q^{i}}}\otimes \mathbf {b} ^{i}}
Note that the contravariant basis vector b i i
b
i
=
∇
ψ
i
{\displaystyle \mathbf {b} ^{i}={\boldsymbol {\nabla }}\psi ^{i}}
Christoffel symbols of the first kind [ edit ] The Christoffel symbols of the first kind are defined as
b
i
,
j
=
∂
b
i
∂
q
j
:=
Γ
i
j
k
b
k
⇒
b
i
,
j
⋅
b
l
=
Γ
i
j
l
{\displaystyle \mathbf {b} _{i,j}={\frac {\partial \mathbf {b} _{i}}{\partial q^{j}}}:=\Gamma _{ijk}~\mathbf {b} ^{k}\quad \Rightarrow \quad \mathbf {b} _{i,j}\cdot \mathbf {b} _{l}=\Gamma _{ijl}}
ijk gij we note that
g
i
j
,
k
=
(
b
i
⋅
b
j
)
,
k
=
b
i
,
k
⋅
b
j
+
b
i
⋅
b
j
,
k
=
Γ
i
k
j
+
Γ
j
k
i
g
i
k
,
j
=
(
b
i
⋅
b
k
)
,
j
=
b
i
,
j
⋅
b
k
+
b
i
⋅
b
k
,
j
=
Γ
i
j
k
+
Γ
k
j
i
g
j
k
,
i
=
(
b
j
⋅
b
k
)
,
i
=
b
j
,
i
⋅
b
k
+
b
j
⋅
b
k
,
i
=
Γ
j
i
k
+
Γ
k
i
j
{\displaystyle {\begin{aligned}g_{ij,k}&=(\mathbf {b} _{i}\cdot \mathbf {b} _{j})_{,k}=\mathbf {b} _{i,k}\cdot \mathbf {b} _{j}+\mathbf {b} _{i}\cdot \mathbf {b} _{j,k}=\Gamma _{ikj}+\Gamma _{jki}\\g_{ik,j}&=(\mathbf {b} _{i}\cdot \mathbf {b} _{k})_{,j}=\mathbf {b} _{i,j}\cdot \mathbf {b} _{k}+\mathbf {b} _{i}\cdot \mathbf {b} _{k,j}=\Gamma _{ijk}+\Gamma _{kji}\\g_{jk,i}&=(\mathbf {b} _{j}\cdot \mathbf {b} _{k})_{,i}=\mathbf {b} _{j,i}\cdot \mathbf {b} _{k}+\mathbf {b} _{j}\cdot \mathbf {b} _{k,i}=\Gamma _{jik}+\Gamma _{kij}\end{aligned}}}
b i,j b j,i ijk jik
Γ
i
j
k
=
1
2
(
g
i
k
,
j
+
g
j
k
,
i
−
g
i
j
,
k
)
=
1
2
[
(
b
i
⋅
b
k
)
,
j
+
(
b
j
⋅
b
k
)
,
i
−
(
b
i
⋅
b
j
)
,
k
]
{\displaystyle \Gamma _{ijk}={\frac {1}{2}}(g_{ik,j}+g_{jk,i}-g_{ij,k})={\frac {1}{2}}[(\mathbf {b} _{i}\cdot \mathbf {b} _{k})_{,j}+(\mathbf {b} _{j}\cdot \mathbf {b} _{k})_{,i}-(\mathbf {b} _{i}\cdot \mathbf {b} _{j})_{,k}]}
Christoffel symbols of the second kind [ edit ] The Christoffel symbols of the second kind are defined as
Γ
i
j
k
=
Γ
j
i
k
{\displaystyle \Gamma _{ij}^{k}=\Gamma _{ji}^{k}}
∂
b
i
∂
q
j
=
Γ
i
j
k
b
k
{\displaystyle {\cfrac {\partial \mathbf {b} _{i}}{\partial q^{j}}}=\Gamma _{ij}^{k}~\mathbf {b} _{k}}
This implies that
Γ
i
j
k
=
∂
b
i
∂
q
j
⋅
b
k
=
−
b
i
⋅
∂
b
k
∂
q
j
{\displaystyle \Gamma _{ij}^{k}={\cfrac {\partial \mathbf {b} _{i}}{\partial q^{j}}}\cdot \mathbf {b} ^{k}=-\mathbf {b} _{i}\cdot {\cfrac {\partial \mathbf {b} ^{k}}{\partial q^{j}}}}
∂
b
i
∂
q
j
=
−
Γ
j
k
i
b
k
;
∇
b
i
=
Γ
i
j
k
b
k
⊗
b
j
;
∇
b
i
=
−
Γ
j
k
i
b
k
⊗
b
j
{\displaystyle {\cfrac {\partial \mathbf {b} ^{i}}{\partial q^{j}}}=-\Gamma _{jk}^{i}~\mathbf {b} ^{k}~;~~{\boldsymbol {\nabla }}\mathbf {b} _{i}=\Gamma _{ij}^{k}~\mathbf {b} _{k}\otimes \mathbf {b} ^{j}~;~~{\boldsymbol {\nabla }}\mathbf {b} ^{i}=-\Gamma _{jk}^{i}~\mathbf {b} ^{k}\otimes \mathbf {b} ^{j}}
Another particularly useful relation, which shows that the Christoffel symbol depends only on the metric tensor and its derivatives, is
Γ
i
j
k
=
g
k
m
2
(
∂
g
m
i
∂
q
j
+
∂
g
m
j
∂
q
i
−
∂
g
i
j
∂
q
m
)
{\displaystyle \Gamma _{ij}^{k}={\frac {g^{km}}{2}}\left({\frac {\partial g_{mi}}{\partial q^{j}}}+{\frac {\partial g_{mj}}{\partial q^{i}}}-{\frac {\partial g_{ij}}{\partial q^{m}}}\right)}
Explicit expression for the gradient of a vector field [ edit ] The following expressions for the gradient of a vector field in curvilinear coordinates are quite useful.
∇
v
=
[
∂
v
i
∂
q
k
+
Γ
l
k
i
v
l
]
b
i
⊗
b
k
=
[
∂
v
i
∂
q
k
−
Γ
k
i
l
v
l
]
b
i
⊗
b
k
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\mathbf {v} &=\left[{\cfrac {\partial v^{i}}{\partial q^{k}}}+\Gamma _{lk}^{i}~v^{l}\right]~\mathbf {b} _{i}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial v_{i}}{\partial q^{k}}}-\Gamma _{ki}^{l}~v_{l}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} ^{k}\end{aligned}}}
Representing a physical vector field [ edit ] The vector field v can be represented as
v
=
v
i
b
i
=
v
^
i
b
^
i
{\displaystyle \mathbf {v} =v_{i}~\mathbf {b} ^{i}={\hat {v}}_{i}~{\hat {\mathbf {b} }}^{i}}
v
i
{\displaystyle v_{i}}
v
^
i
{\displaystyle {\hat {v}}_{i}}
summation )
b
^
i
=
b
i
g
i
i
{\displaystyle {\hat {\mathbf {b} }}^{i}={\cfrac {\mathbf {b} ^{i}}{\sqrt {g^{ii}}}}}
Second-order tensor field [ edit ] The gradient of a second order tensor field can similarly be expressed as
∇
S
=
∂
S
∂
q
i
⊗
b
i
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {S}}={\frac {\partial {\boldsymbol {S}}}{\partial q^{i}}}\otimes \mathbf {b} ^{i}}
Explicit expressions for the gradient [ edit ] If we consider the expression for the tensor in terms of a contravariant basis, then
∇
S
=
∂
∂
q
k
[
S
i
j
b
i
⊗
b
j
]
⊗
b
k
=
[
∂
S
i
j
∂
q
k
−
Γ
k
i
l
S
l
j
−
Γ
k
j
l
S
i
l
]
b
i
⊗
b
j
⊗
b
k
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {S}}={\frac {\partial }{\partial q^{k}}}[S_{ij}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}]\otimes \mathbf {b} ^{k}=\left[{\frac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}\otimes \mathbf {b} ^{k}}
∇
S
=
[
∂
S
i
j
∂
q
k
+
Γ
k
l
i
S
l
j
+
Γ
k
l
j
S
i
l
]
b
i
⊗
b
j
⊗
b
k
=
[
∂
S
j
i
∂
q
k
+
Γ
k
l
i
S
j
l
−
Γ
k
j
l
S
l
i
]
b
i
⊗
b
j
⊗
b
k
=
[
∂
S
i
j
∂
q
k
−
Γ
i
k
l
S
l
j
+
Γ
k
l
j
S
i
l
]
b
i
⊗
b
j
⊗
b
k
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}{\boldsymbol {S}}&=\left[{\cfrac {\partial S^{ij}}{\partial q^{k}}}+\Gamma _{kl}^{i}~S^{lj}+\Gamma _{kl}^{j}~S^{il}\right]~\mathbf {b} _{i}\otimes \mathbf {b} _{j}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{k}}}+\Gamma _{kl}^{i}~S_{~j}^{l}-\Gamma _{kj}^{l}~S_{~l}^{i}\right]~\mathbf {b} _{i}\otimes \mathbf {b} ^{j}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} _{j}\otimes \mathbf {b} ^{k}\end{aligned}}}
Representing a physical second-order tensor field [ edit ] The physical components of a second-order tensor field can be obtained by using a normalized contravariant basis, i.e.,
S
=
S
i
j
b
i
⊗
b
j
=
S
^
i
j
b
^
i
⊗
b
^
j
{\displaystyle {\boldsymbol {S}}=S_{ij}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}={\hat {S}}_{ij}~{\hat {\mathbf {b} }}^{i}\otimes {\hat {\mathbf {b} }}^{j}}
S
^
i
j
=
S
i
j
g
i
i
g
j
j
{\displaystyle {\hat {S}}_{ij}=S_{ij}~{\sqrt {g^{ii}~g^{jj}}}}
The divergence of a vector field (
v
{\displaystyle \mathbf {v} }
div
v
=
∇
⋅
v
=
tr
(
∇
v
)
{\displaystyle \operatorname {div} ~\mathbf {v} ={\boldsymbol {\nabla }}\cdot \mathbf {v} ={\text{tr}}({\boldsymbol {\nabla }}\mathbf {v} )}
∇
⋅
v
=
∂
v
i
∂
q
i
+
Γ
ℓ
i
i
v
ℓ
=
[
∂
v
i
∂
q
j
−
Γ
j
i
ℓ
v
ℓ
]
g
i
j
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {\partial v^{i}}{\partial q^{i}}}+\Gamma _{\ell i}^{i}~v^{\ell }=\left[{\cfrac {\partial v_{i}}{\partial q^{j}}}-\Gamma _{ji}^{\ell }~v_{\ell }\right]~g^{ij}}
An alternative equation for the divergence of a vector field is frequently used. To derive this relation recall that
∇
⋅
v
=
∂
v
i
∂
q
i
+
Γ
ℓ
i
i
v
ℓ
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\frac {\partial v^{i}}{\partial q^{i}}}+\Gamma _{\ell i}^{i}~v^{\ell }}
Γ
ℓ
i
i
=
Γ
i
ℓ
i
=
g
m
i
2
[
∂
g
i
m
∂
q
ℓ
+
∂
g
ℓ
m
∂
q
i
−
∂
g
i
l
∂
q
m
]
{\displaystyle \Gamma _{\ell i}^{i}=\Gamma _{i\ell }^{i}={\cfrac {g^{mi}}{2}}\left[{\frac {\partial g_{im}}{\partial q^{\ell }}}+{\frac {\partial g_{\ell m}}{\partial q^{i}}}-{\frac {\partial g_{il}}{\partial q^{m}}}\right]}
g
{\displaystyle {\boldsymbol {g}}}
g
m
i
∂
g
ℓ
m
∂
q
i
=
g
m
i
∂
g
i
ℓ
∂
q
m
{\displaystyle g^{mi}~{\frac {\partial g_{\ell m}}{\partial q^{i}}}=g^{mi}~{\frac {\partial g_{i\ell }}{\partial q^{m}}}}
∇
⋅
v
=
∂
v
i
∂
q
i
+
g
m
i
2
∂
g
i
m
∂
q
ℓ
v
ℓ
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\frac {\partial v^{i}}{\partial q^{i}}}+{\cfrac {g^{mi}}{2}}~{\frac {\partial g_{im}}{\partial q^{\ell }}}~v^{\ell }}
gij ] is the matrix whose components are gij , then the inverse of the matrix is
[
g
i
j
]
−
1
=
[
g
i
j
]
{\displaystyle [g_{ij}]^{-1}=[g^{ij}]}
[
g
i
j
]
=
[
g
i
j
]
−
1
=
A
i
j
g
;
g
:=
det
(
[
g
i
j
]
)
=
det
g
{\displaystyle [g^{ij}]=[g_{ij}]^{-1}={\cfrac {A^{ij}}{g}}~;~~g:=\det([g_{ij}])=\det {\boldsymbol {g}}}
Aij are the Cofactor matrix of the components gij . From matrix algebra we have
g
=
det
(
[
g
i
j
]
)
=
∑
i
g
i
j
A
i
j
⇒
∂
g
∂
g
i
j
=
A
i
j
{\displaystyle g=\det([g_{ij}])=\sum _{i}g_{ij}~A^{ij}\quad \Rightarrow \quad {\frac {\partial g}{\partial g_{ij}}}=A^{ij}}
[
g
i
j
]
=
1
g
∂
g
∂
g
i
j
{\displaystyle [g^{ij}]={\cfrac {1}{g}}~{\frac {\partial g}{\partial g_{ij}}}}
∇
⋅
v
=
∂
v
i
∂
q
i
+
1
2
g
∂
g
∂
g
m
i
∂
g
i
m
∂
q
ℓ
v
ℓ
=
∂
v
i
∂
q
i
+
1
2
g
∂
g
∂
q
ℓ
v
ℓ
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\frac {\partial v^{i}}{\partial q^{i}}}+{\cfrac {1}{2g}}~{\frac {\partial g}{\partial g_{mi}}}~{\frac {\partial g_{im}}{\partial q^{\ell }}}~v^{\ell }={\frac {\partial v^{i}}{\partial q^{i}}}+{\cfrac {1}{2g}}~{\frac {\partial g}{\partial q^{\ell }}}~v^{\ell }}
∇
⋅
v
=
1
g
∂
∂
q
i
(
v
i
g
)
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {1}{\sqrt {g}}}~{\frac {\partial }{\partial q^{i}}}(v^{i}~{\sqrt {g}})}
Second-order tensor field [ edit ] The divergence of a second-order tensor field is defined using
(
∇
⋅
S
)
⋅
a
=
∇
⋅
(
S
a
)
{\displaystyle ({\boldsymbol {\nabla }}\cdot {\boldsymbol {S}})\cdot \mathbf {a} ={\boldsymbol {\nabla }}\cdot ({\boldsymbol {S}}\mathbf {a} )}
a is an arbitrary constant vector.
[ 11]
∇
⋅
S
=
[
∂
S
i
j
∂
q
k
−
Γ
k
i
l
S
l
j
−
Γ
k
j
l
S
i
l
]
g
i
k
b
j
=
[
∂
S
i
j
∂
q
i
+
Γ
i
l
i
S
l
j
+
Γ
i
l
j
S
i
l
]
b
j
=
[
∂
S
j
i
∂
q
i
+
Γ
i
l
i
S
j
l
−
Γ
i
j
l
S
l
i
]
b
j
=
[
∂
S
i
j
∂
q
k
−
Γ
i
k
l
S
l
j
+
Γ
k
l
j
S
i
l
]
g
i
k
b
j
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left[{\cfrac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~g^{ik}~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S^{ij}}{\partial q^{i}}}+\Gamma _{il}^{i}~S^{lj}+\Gamma _{il}^{j}~S^{il}\right]~\mathbf {b} _{j}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{i}}}+\Gamma _{il}^{i}~S_{~j}^{l}-\Gamma _{ij}^{l}~S_{~l}^{i}\right]~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~g^{ik}~\mathbf {b} _{j}\end{aligned}}}
The Laplacian of a scalar field φ(x ) is defined as
∇
2
φ
:=
∇
⋅
(
∇
φ
)
{\displaystyle \nabla ^{2}\varphi :={\boldsymbol {\nabla }}\cdot ({\boldsymbol {\nabla }}\varphi )}
∇
2
φ
=
1
g
∂
∂
q
i
(
[
∇
φ
]
i
g
)
{\displaystyle \nabla ^{2}\varphi ={\cfrac {1}{\sqrt {g}}}~{\frac {\partial }{\partial q^{i}}}([{\boldsymbol {\nabla }}\varphi ]^{i}~{\sqrt {g}})}
∇
φ
=
∂
φ
∂
q
l
b
l
=
g
l
i
∂
φ
∂
q
l
b
i
⇒
[
∇
φ
]
i
=
g
l
i
∂
φ
∂
q
l
{\displaystyle {\boldsymbol {\nabla }}\varphi ={\frac {\partial \varphi }{\partial q^{l}}}~\mathbf {b} ^{l}=g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}~\mathbf {b} _{i}\quad \Rightarrow \quad [{\boldsymbol {\nabla }}\varphi ]^{i}=g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}}
∇
2
φ
=
1
g
∂
∂
q
i
(
g
l
i
∂
φ
∂
q
l
g
)
{\displaystyle \nabla ^{2}\varphi ={\cfrac {1}{\sqrt {g}}}~{\frac {\partial }{\partial q^{i}}}\left(g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}~{\sqrt {g}}\right)}
Curl of a vector field [ edit ] The curl of a vector field v in covariant curvilinear coordinates can be written as
∇
×
v
=
E
r
s
t
v
s
|
r
b
t
{\displaystyle {\boldsymbol {\nabla }}\times \mathbf {v} ={\mathcal {E}}^{rst}v_{s|r}~\mathbf {b} _{t}}
v
s
|
r
=
v
s
,
r
−
Γ
s
r
i
v
i
{\displaystyle v_{s|r}=v_{s,r}-\Gamma _{sr}^{i}~v_{i}}
Orthogonal curvilinear coordinates [ edit ] Assume, for the purposes of this section, that the curvilinear coordinate system is orthogonal , i.e.,
b
i
⋅
b
j
=
{
g
i
i
if
i
=
j
0
if
i
≠
j
,
{\displaystyle \mathbf {b} _{i}\cdot \mathbf {b} _{j}={\begin{cases}g_{ii}&{\text{if }}i=j\\0&{\text{if }}i\neq j,\end{cases}}}
b
i
⋅
b
j
=
{
g
i
i
if
i
=
j
0
if
i
≠
j
,
{\displaystyle \mathbf {b} ^{i}\cdot \mathbf {b} ^{j}={\begin{cases}g^{ii}&{\text{if }}i=j\\0&{\text{if }}i\neq j,\end{cases}}}
g
i
i
=
g
i
i
−
1
{\displaystyle g^{ii}=g_{ii}^{-1}}
b
i
,
b
j
{\displaystyle \mathbf {b} _{i},\mathbf {b} _{j}}
b i b j e 1 , e 2 , e 3 ) be a background, fixed, Cartesian basis. A list of orthogonal curvilinear coordinates is given below.
Metric tensor in orthogonal curvilinear coordinates [ edit ] Let r (x ) be the position vector of the point x with respect to the origin of the coordinate system. The notation can be simplified by noting that x = r (x ). At each point we can construct a small line element dx . The square of the length of the line element is the scalar product dx • dx and is called the metric of the space . Recall that the space of interest is assumed to be Euclidean when we talk of curvilinear coordinates. Let us express the position vector in terms of the background, fixed, Cartesian basis, i.e.,
x
=
∑
i
=
1
3
x
i
e
i
{\displaystyle \mathbf {x} =\sum _{i=1}^{3}x_{i}~\mathbf {e} _{i}}
Using the chain rule , we can then express dx in terms of three-dimensional orthogonal curvilinear coordinates (q 1 , q 2 , q 3 ) as
d
x
=
∑
i
=
1
3
∑
j
=
1
3
(
∂
x
i
∂
q
j
e
i
)
d
q
j
{\displaystyle \mathrm {d} \mathbf {x} =\sum _{i=1}^{3}\sum _{j=1}^{3}\left({\cfrac {\partial x_{i}}{\partial q^{j}}}~\mathbf {e} _{i}\right)\mathrm {d} q^{j}}
d
x
⋅
d
x
=
∑
i
=
1
3
∑
j
=
1
3
∑
k
=
1
3
∂
x
i
∂
q
j
∂
x
i
∂
q
k
d
q
j
d
q
k
{\displaystyle \mathrm {d} \mathbf {x} \cdot \mathrm {d} \mathbf {x} =\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}{\cfrac {\partial x_{i}}{\partial q^{j}}}~{\cfrac {\partial x_{i}}{\partial q^{k}}}~\mathrm {d} q^{j}~\mathrm {d} q^{k}}
The symmetric quantity
g
i
j
(
q
i
,
q
j
)
=
∑
k
=
1
3
∂
x
k
∂
q
i
∂
x
k
∂
q
j
=
b
i
⋅
b
j
{\displaystyle g_{ij}(q^{i},q^{j})=\sum _{k=1}^{3}{\cfrac {\partial x_{k}}{\partial q^{i}}}~{\cfrac {\partial x_{k}}{\partial q^{j}}}=\mathbf {b} _{i}\cdot \mathbf {b} _{j}}
fundamental (or metric) tensor of the Euclidean space in curvilinear coordinates.
Note also that
g
i
j
=
∂
x
∂
q
i
⋅
∂
x
∂
q
j
=
(
∑
k
h
k
i
e
k
)
⋅
(
∑
m
h
m
j
e
m
)
=
∑
k
h
k
i
h
k
j
{\displaystyle g_{ij}={\cfrac {\partial \mathbf {x} }{\partial q^{i}}}\cdot {\cfrac {\partial \mathbf {x} }{\partial q^{j}}}=\left(\sum _{k}h_{ki}~\mathbf {e} _{k}\right)\cdot \left(\sum _{m}h_{mj}~\mathbf {e} _{m}\right)=\sum _{k}h_{ki}~h_{kj}}
hij are the Lamé coefficients.
If we define the scale factors, hi , using
b
i
⋅
b
i
=
g
i
i
=
∑
k
h
k
i
2
=:
h
i
2
⇒
|
∂
x
∂
q
i
|
=
|
b
i
|
=
g
i
i
=
h
i
{\displaystyle \mathbf {b} _{i}\cdot \mathbf {b} _{i}=g_{ii}=\sum _{k}h_{ki}^{2}=:h_{i}^{2}\quad \Rightarrow \quad \left|{\cfrac {\partial \mathbf {x} }{\partial q^{i}}}\right|=\left|\mathbf {b} _{i}\right|={\sqrt {g_{ii}}}=h_{i}}
Example: Polar coordinates [ edit ] If we consider polar coordinates for R 2 , note that
(
x
,
y
)
=
(
r
cos
θ
,
r
sin
θ
)
{\displaystyle (x,y)=(r\cos \theta ,r\sin \theta )}
r ,θ) → (r cos θ, r sin θ) is r .
The orthogonal basis vectors are b r b θ = (−r sin θ, r cos θ). The normalized basis vectors are e r e θ = (−sin θ, cos θ) and the scale factors are h r h θ = r . The fundamental tensor is g 11 =1, g 22 =r 2 , g 12 = g 21 =0.
Line and surface integrals [ edit ] If we wish to use curvilinear coordinates for vector calculus calculations, adjustments need to be made in the calculation of line, surface and volume integrals. For simplicity, we again restrict the discussion to three dimensions and orthogonal curvilinear coordinates. However, the same arguments apply for
n
{\displaystyle n}
Normally in the calculation of line integrals we are interested in calculating
∫
C
f
d
s
=
∫
a
b
f
(
x
(
t
)
)
|
∂
x
∂
t
|
d
t
{\displaystyle \int _{C}f\,ds=\int _{a}^{b}f(\mathbf {x} (t))\left|{\partial \mathbf {x} \over \partial t}\right|\;dt}
x t ) parametrizes C in Cartesian coordinates.
In curvilinear coordinates, the term
|
∂
x
∂
t
|
=
|
∑
i
=
1
3
∂
x
∂
q
i
∂
q
i
∂
t
|
{\displaystyle \left|{\partial \mathbf {x} \over \partial t}\right|=\left|\sum _{i=1}^{3}{\partial \mathbf {x} \over \partial q^{i}}{\partial q^{i} \over \partial t}\right|}
by the chain rule . And from the definition of the Lamé coefficients,
∂
x
∂
q
i
=
∑
k
h
k
i
e
k
{\displaystyle {\partial \mathbf {x} \over \partial q^{i}}=\sum _{k}h_{ki}~\mathbf {e} _{k}}
and thus
|
∂
x
∂
t
|
=
|
∑
k
(
∑
i
h
k
i
∂
q
i
∂
t
)
e
k
|
=
∑
i
∑
j
∑
k
h
k
i
h
k
j
∂
q
i
∂
t
∂
q
j
∂
t
=
∑
i
∑
j
g
i
j
∂
q
i
∂
t
∂
q
j
∂
t
{\displaystyle {\begin{aligned}\left|{\partial \mathbf {x} \over \partial t}\right|&=\left|\sum _{k}\left(\sum _{i}h_{ki}~{\cfrac {\partial q^{i}}{\partial t}}\right)\mathbf {e} _{k}\right|\\[8pt]&={\sqrt {\sum _{i}\sum _{j}\sum _{k}h_{ki}~h_{kj}{\cfrac {\partial q^{i}}{\partial t}}{\cfrac {\partial q^{j}}{\partial t}}}}={\sqrt {\sum _{i}\sum _{j}g_{ij}~{\cfrac {\partial q^{i}}{\partial t}}{\cfrac {\partial q^{j}}{\partial t}}}}\end{aligned}}}
Now, since
g
i
j
=
0
{\displaystyle g_{ij}=0}
i
≠
j
{\displaystyle i\neq j}
|
∂
x
∂
t
|
=
∑
i
g
i
i
(
∂
q
i
∂
t
)
2
=
∑
i
h
i
2
(
∂
q
i
∂
t
)
2
{\displaystyle \left|{\partial \mathbf {x} \over \partial t}\right|={\sqrt {\sum _{i}g_{ii}~\left({\cfrac {\partial q^{i}}{\partial t}}\right)^{2}}}={\sqrt {\sum _{i}h_{i}^{2}~\left({\cfrac {\partial q^{i}}{\partial t}}\right)^{2}}}}
Likewise, if we are interested in a surface integral , the relevant calculation, with the parameterization of the surface in Cartesian coordinates is:
∫
S
f
d
S
=
∬
T
f
(
x
(
s
,
t
)
)
|
∂
x
∂
s
×
∂
x
∂
t
|
d
s
d
t
{\displaystyle \int _{S}f\,dS=\iint _{T}f(\mathbf {x} (s,t))\left|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right|\,ds\,dt}
|
∂
x
∂
s
×
∂
x
∂
t
|
=
|
(
∑
i
∂
x
∂
q
i
∂
q
i
∂
s
)
×
(
∑
j
∂
x
∂
q
j
∂
q
j
∂
t
)
|
{\displaystyle \left|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right|=\left|\left(\sum _{i}{\partial \mathbf {x} \over \partial q^{i}}{\partial q^{i} \over \partial s}\right)\times \left(\sum _{j}{\partial \mathbf {x} \over \partial q^{j}}{\partial q^{j} \over \partial t}\right)\right|}
∂
x
∂
q
i
∂
q
i
∂
s
=
∑
k
(
∑
i
=
1
3
h
k
i
∂
q
i
∂
s
)
e
k
;
∂
x
∂
q
j
∂
q
j
∂
t
=
∑
m
(
∑
j
=
1
3
h
m
j
∂
q
j
∂
t
)
e
m
{\displaystyle {\partial \mathbf {x} \over \partial q^{i}}{\partial q^{i} \over \partial s}=\sum _{k}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\mathbf {e} _{k}~;~~{\partial \mathbf {x} \over \partial q^{j}}{\partial q^{j} \over \partial t}=\sum _{m}\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{m}}
Therefore,
|
∂
x
∂
s
×
∂
x
∂
t
|
=
|
∑
k
∑
m
(
∑
i
=
1
3
h
k
i
∂
q
i
∂
s
)
(
∑
j
=
1
3
h
m
j
∂
q
j
∂
t
)
e
k
×
e
m
|
=
|
∑
p
∑
k
∑
m
E
k
m
p
(
∑
i
=
1
3
h
k
i
∂
q
i
∂
s
)
(
∑
j
=
1
3
h
m
j
∂
q
j
∂
t
)
e
p
|
{\displaystyle {\begin{aligned}\left|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right|&=\left|\sum _{k}\sum _{m}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{k}\times \mathbf {e} _{m}\right|\\[8pt]&=\left|\sum _{p}\sum _{k}\sum _{m}{\mathcal {E}}_{kmp}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{p}\right|\end{aligned}}}
E
{\displaystyle {\mathcal {E}}}
permutation symbol .
In determinant form, the cross product in terms of curvilinear coordinates will be:
|
e
1
e
2
e
3
∑
i
h
1
i
∂
q
i
∂
s
∑
i
h
2
i
∂
q
i
∂
s
∑
i
h
3
i
∂
q
i
∂
s
∑
j
h
1
j
∂
q
j
∂
t
∑
j
h
2
j
∂
q
j
∂
t
∑
j
h
3
j
∂
q
j
∂
t
|
{\displaystyle {\begin{vmatrix}\mathbf {e} _{1}&\mathbf {e} _{2}&\mathbf {e} _{3}\\&&\\\sum _{i}h_{1i}{\partial q^{i} \over \partial s}&\sum _{i}h_{2i}{\partial q^{i} \over \partial s}&\sum _{i}h_{3i}{\partial q^{i} \over \partial s}\\&&\\\sum _{j}h_{1j}{\partial q^{j} \over \partial t}&\sum _{j}h_{2j}{\partial q^{j} \over \partial t}&\sum _{j}h_{3j}{\partial q^{j} \over \partial t}\end{vmatrix}}}
[ edit ] In orthogonal curvilinear coordinates of 3 dimensions, where
b
i
=
∑
k
g
i
k
b
k
;
g
i
i
=
1
g
i
i
=
1
h
i
2
{\displaystyle \mathbf {b} ^{i}=\sum _{k}g^{ik}~\mathbf {b} _{k}~;~~g^{ii}={\cfrac {1}{g_{ii}}}={\cfrac {1}{h_{i}^{2}}}}
gradient of a scalar or vector field as
∇
φ
=
∑
i
∂
φ
∂
q
i
b
i
=
∑
i
∑
j
∂
φ
∂
q
i
g
i
j
b
j
=
∑
i
1
h
i
2
∂
f
∂
q
i
b
i
;
∇
v
=
∑
i
1
h
i
2
∂
v
∂
q
i
⊗
b
i
{\displaystyle \nabla \varphi =\sum _{i}{\partial \varphi \over \partial q^{i}}~\mathbf {b} ^{i}=\sum _{i}\sum _{j}{\partial \varphi \over \partial q^{i}}~g^{ij}~\mathbf {b} _{j}=\sum _{i}{\cfrac {1}{h_{i}^{2}}}~{\partial f \over \partial q^{i}}~\mathbf {b} _{i}~;~~\nabla \mathbf {v} =\sum _{i}{\cfrac {1}{h_{i}^{2}}}~{\partial \mathbf {v} \over \partial q^{i}}\otimes \mathbf {b} _{i}}
g
=
g
11
g
22
g
33
=
h
1
2
h
2
2
h
3
2
⇒
g
=
h
1
h
2
h
3
{\displaystyle g=g_{11}~g_{22}~g_{33}=h_{1}^{2}~h_{2}^{2}~h_{3}^{2}\quad \Rightarrow \quad {\sqrt {g}}=h_{1}h_{2}h_{3}}
divergence of a vector field can then be written as
∇
⋅
v
=
1
h
1
h
2
h
3
∂
∂
q
i
(
h
1
h
2
h
3
v
i
)
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {1}{h_{1}h_{2}h_{3}}}~{\frac {\partial }{\partial q^{i}}}(h_{1}h_{2}h_{3}~v^{i})}
v
i
=
g
i
k
v
k
⇒
v
1
=
g
11
v
1
=
v
1
h
1
2
;
v
2
=
g
22
v
2
=
v
2
h
2
2
;
v
3
=
g
33
v
3
=
v
3
h
3
2
{\displaystyle v^{i}=g^{ik}~v_{k}\quad \Rightarrow v^{1}=g^{11}~v_{1}={\cfrac {v_{1}}{h_{1}^{2}}}~;~~v^{2}=g^{22}~v_{2}={\cfrac {v_{2}}{h_{2}^{2}}}~;~~v^{3}=g^{33}~v_{3}={\cfrac {v_{3}}{h_{3}^{2}}}}
∇
⋅
v
=
1
h
1
h
2
h
3
∑
i
∂
∂
q
i
(
h
1
h
2
h
3
h
i
2
v
i
)
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {1}{h_{1}h_{2}h_{3}}}~\sum _{i}{\frac {\partial }{\partial q^{i}}}\left({\cfrac {h_{1}h_{2}h_{3}}{h_{i}^{2}}}~v_{i}\right)}
Laplacian in a similar manner by noting that
g
l
i
∂
φ
∂
q
l
=
{
g
11
∂
φ
∂
q
1
,
g
22
∂
φ
∂
q
2
,
g
33
∂
φ
∂
q
3
}
=
{
1
h
1
2
∂
φ
∂
q
1
,
1
h
2
2
∂
φ
∂
q
2
,
1
h
3
2
∂
φ
∂
q
3
}
{\displaystyle g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}=\left\{g^{11}~{\frac {\partial \varphi }{\partial q^{1}}},g^{22}~{\frac {\partial \varphi }{\partial q^{2}}},g^{33}~{\frac {\partial \varphi }{\partial q^{3}}}\right\}=\left\{{\cfrac {1}{h_{1}^{2}}}~{\frac {\partial \varphi }{\partial q^{1}}},{\cfrac {1}{h_{2}^{2}}}~{\frac {\partial \varphi }{\partial q^{2}}},{\cfrac {1}{h_{3}^{2}}}~{\frac {\partial \varphi }{\partial q^{3}}}\right\}}
∇
2
φ
=
1
h
1
h
2
h
3
∑
i
∂
∂
q
i
(
h
1
h
2
h
3
h
i
2
∂
φ
∂
q
i
)
{\displaystyle \nabla ^{2}\varphi ={\cfrac {1}{h_{1}h_{2}h_{3}}}~\sum _{i}{\frac {\partial }{\partial q^{i}}}\left({\cfrac {h_{1}h_{2}h_{3}}{h_{i}^{2}}}~{\frac {\partial \varphi }{\partial q^{i}}}\right)}
n -dimensions.
The curl of a vector field is given by
∇
×
v
=
1
h
1
h
2
h
3
∑
i
=
1
n
e
i
∑
j
k
ε
i
j
k
h
i
∂
(
h
k
v
k
)
∂
q
j
{\displaystyle \nabla \times \mathbf {v} ={\frac {1}{h_{1}h_{2}h_{3}}}\sum _{i=1}^{n}\mathbf {e} _{i}\sum _{jk}\varepsilon _{ijk}h_{i}{\frac {\partial (h_{k}v_{k})}{\partial q^{j}}}}
ijk Levi-Civita symbol .
Example: Cylindrical polar coordinates [ edit ] For cylindrical coordinates we have
(
x
1
,
x
2
,
x
3
)
=
x
=
φ
(
q
1
,
q
2
,
q
3
)
=
φ
(
r
,
θ
,
z
)
=
{
r
cos
θ
,
r
sin
θ
,
z
}
{\displaystyle (x_{1},x_{2},x_{3})=\mathbf {x} ={\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})={\boldsymbol {\varphi }}(r,\theta ,z)=\{r\cos \theta ,r\sin \theta ,z\}}
{
ψ
1
(
x
)
,
ψ
2
(
x
)
,
ψ
3
(
x
)
}
=
(
q
1
,
q
2
,
q
3
)
≡
(
r
,
θ
,
z
)
=
{
x
1
2
+
x
2
2
,
tan
−
1
(
x
2
/
x
1
)
,
x
3
}
{\displaystyle \{\psi ^{1}(\mathbf {x} ),\psi ^{2}(\mathbf {x} ),\psi ^{3}(\mathbf {x} )\}=(q^{1},q^{2},q^{3})\equiv (r,\theta ,z)=\{{\sqrt {x_{1}^{2}+x_{2}^{2}}},\tan ^{-1}(x_{2}/x_{1}),x_{3}\}}
0
<
r
<
∞
,
0
<
θ
<
2
π
,
−
∞
<
z
<
∞
{\displaystyle 0<r<\infty ~,~~0<\theta <2\pi ~,~~-\infty <z<\infty }
Then the covariant and contravariant basis vectors are
b
1
=
e
r
=
b
1
b
2
=
r
e
θ
=
r
2
b
2
b
3
=
e
z
=
b
3
{\displaystyle {\begin{aligned}\mathbf {b} _{1}&=\mathbf {e} _{r}=\mathbf {b} ^{1}\\\mathbf {b} _{2}&=r~\mathbf {e} _{\theta }=r^{2}~\mathbf {b} ^{2}\\\mathbf {b} _{3}&=\mathbf {e} _{z}=\mathbf {b} ^{3}\end{aligned}}}
e
r
,
e
θ
,
e
z
{\displaystyle \mathbf {e} _{r},\mathbf {e} _{\theta },\mathbf {e} _{z}}
r
,
θ
,
z
{\displaystyle r,\theta ,z}
Note that the components of the metric tensor are such that
g
i
j
=
g
i
j
=
0
(
i
≠
j
)
;
g
11
=
1
,
g
22
=
1
r
,
g
33
=
1
{\displaystyle g^{ij}=g_{ij}=0(i\neq j)~;~~{\sqrt {g^{11}}}=1,~{\sqrt {g^{22}}}={\cfrac {1}{r}},~{\sqrt {g^{33}}}=1}
The non-zero components of the Christoffel symbol of the second kind are
Γ
12
2
=
Γ
21
2
=
1
r
;
Γ
22
1
=
−
r
{\displaystyle \Gamma _{12}^{2}=\Gamma _{21}^{2}={\cfrac {1}{r}}~;~~\Gamma _{22}^{1}=-r}
Representing a physical vector field [ edit ] The normalized contravariant basis vectors in cylindrical polar coordinates are
b
^
1
=
e
r
;
b
^
2
=
e
θ
;
b
^
3
=
e
z
{\displaystyle {\hat {\mathbf {b} }}^{1}=\mathbf {e} _{r}~;~~{\hat {\mathbf {b} }}^{2}=\mathbf {e} _{\theta }~;~~{\hat {\mathbf {b} }}^{3}=\mathbf {e} _{z}}
v are
(
v
^
1
,
v
^
2
,
v
^
3
)
=
(
v
1
,
v
2
/
r
,
v
3
)
=:
(
v
r
,
v
θ
,
v
z
)
{\displaystyle ({\hat {v}}_{1},{\hat {v}}_{2},{\hat {v}}_{3})=(v_{1},v_{2}/r,v_{3})=:(v_{r},v_{\theta },v_{z})}
Gradient of a scalar field [ edit ] The gradient of a scalar field, f (x ), in cylindrical coordinates can now be computed from the general expression in curvilinear coordinates and has the form
∇
f
=
∂
f
∂
r
e
r
+
1
r
∂
f
∂
θ
e
θ
+
∂
f
∂
z
e
z
{\displaystyle {\boldsymbol {\nabla }}f={\cfrac {\partial f}{\partial r}}~\mathbf {e} _{r}+{\cfrac {1}{r}}~{\cfrac {\partial f}{\partial \theta }}~\mathbf {e} _{\theta }+{\cfrac {\partial f}{\partial z}}~\mathbf {e} _{z}}
Gradient of a vector field [ edit ] Similarly, the gradient of a vector field, v (x ), in cylindrical coordinates can be shown to be
∇
v
=
∂
v
r
∂
r
e
r
⊗
e
r
+
1
r
(
∂
v
r
∂
θ
−
v
θ
)
e
r
⊗
e
θ
+
∂
v
r
∂
z
e
r
⊗
e
z
+
∂
v
θ
∂
r
e
θ
⊗
e
r
+
1
r
(
∂
v
θ
∂
θ
+
v
r
)
e
θ
⊗
e
θ
+
∂
v
θ
∂
z
e
θ
⊗
e
z
+
∂
v
z
∂
r
e
z
⊗
e
r
+
1
r
∂
v
z
∂
θ
e
z
⊗
e
θ
+
∂
v
z
∂
z
e
z
⊗
e
z
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{r}}{\partial \theta }}-v_{\theta }\right)~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{r}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\cfrac {\partial v_{\theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{\theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\cfrac {\partial v_{z}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}{\cfrac {\partial v_{z}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{z}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}}
Divergence of a vector field [ edit ] Using the equation for the divergence of a vector field in curvilinear coordinates, the divergence in cylindrical coordinates can be shown to be
∇
⋅
v
=
∂
v
r
∂
r
+
1
r
(
∂
v
θ
∂
θ
+
v
r
)
+
∂
v
z
∂
z
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)+{\cfrac {\partial v_{z}}{\partial z}}\end{aligned}}}
Laplacian of a scalar field [ edit ] The Laplacian is more easily computed by noting that
∇
2
f
=
∇
⋅
∇
f
{\displaystyle {\boldsymbol {\nabla }}^{2}f={\boldsymbol {\nabla }}\cdot {\boldsymbol {\nabla }}f}
v
=
∇
f
=
[
v
r
v
θ
v
z
]
=
[
∂
f
∂
r
1
r
∂
f
∂
θ
∂
f
∂
z
]
{\displaystyle \mathbf {v} ={\boldsymbol {\nabla }}f=\left[v_{r}~~v_{\theta }~~v_{z}\right]=\left[{\cfrac {\partial f}{\partial r}}~~{\cfrac {1}{r}}{\cfrac {\partial f}{\partial \theta }}~~{\cfrac {\partial f}{\partial z}}\right]}
∇
⋅
v
=
∇
2
f
=
∂
2
f
∂
r
2
+
1
r
(
1
r
∂
2
f
∂
θ
2
+
∂
f
∂
r
)
+
∂
2
f
∂
z
2
=
1
r
[
∂
∂
r
(
r
∂
f
∂
r
)
]
+
1
r
2
∂
2
f
∂
θ
2
+
∂
2
f
∂
z
2
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\boldsymbol {\nabla }}^{2}f={\cfrac {\partial ^{2}f}{\partial r^{2}}}+{\cfrac {1}{r}}\left({\cfrac {1}{r}}{\cfrac {\partial ^{2}f}{\partial \theta ^{2}}}+{\cfrac {\partial f}{\partial r}}\right)+{\cfrac {\partial ^{2}f}{\partial z^{2}}}={\cfrac {1}{r}}\left[{\cfrac {\partial }{\partial r}}\left(r{\cfrac {\partial f}{\partial r}}\right)\right]+{\cfrac {1}{r^{2}}}{\cfrac {\partial ^{2}f}{\partial \theta ^{2}}}+{\cfrac {\partial ^{2}f}{\partial z^{2}}}}
Representing a physical second-order tensor field [ edit ] The physical components of a second-order tensor field are those obtained when the tensor is expressed in terms of a normalized contravariant basis. In cylindrical polar coordinates these components are:
S
^
11
=
S
11
=:
S
r
r
,
S
^
12
=
S
12
r
=:
S
r
θ
,
S
^
13
=
S
13
=:
S
r
z
S
^
21
=
S
21
r
=:
S
θ
r
,
S
^
22
=
S
22
r
2
=:
S
θ
θ
,
S
^
23
=
S
23
r
=:
S
θ
z
S
^
31
=
S
31
=:
S
z
r
,
S
^
32
=
S
32
r
=:
S
z
θ
,
S
^
33
=
S
33
=:
S
z
z
{\displaystyle {\begin{aligned}{\hat {S}}_{11}&=S_{11}=:S_{rr},&{\hat {S}}_{12}&={\frac {S_{12}}{r}}=:S_{r\theta },&{\hat {S}}_{13}&=S_{13}=:S_{rz}\\[6pt]{\hat {S}}_{21}&={\frac {S_{21}}{r}}=:S_{\theta r},&{\hat {S}}_{22}&={\frac {S_{22}}{r^{2}}}=:S_{\theta \theta },&{\hat {S}}_{23}&={\frac {S_{23}}{r}}=:S_{\theta z}\\[6pt]{\hat {S}}_{31}&=S_{31}=:S_{zr},&{\hat {S}}_{32}&={\frac {S_{32}}{r}}=:S_{z\theta },&{\hat {S}}_{33}&=S_{33}=:S_{zz}\end{aligned}}}
Gradient of a second-order tensor field [ edit ] Using the above definitions we can show that the gradient of a second-order tensor field in cylindrical polar coordinates can be expressed as
∇
S
=
∂
S
r
r
∂
r
e
r
⊗
e
r
⊗
e
r
+
1
r
[
∂
S
r
r
∂
θ
−
(
S
θ
r
+
S
r
θ
)
]
e
r
⊗
e
r
⊗
e
θ
+
∂
S
r
r
∂
z
e
r
⊗
e
r
⊗
e
z
+
∂
S
r
θ
∂
r
e
r
⊗
e
θ
⊗
e
r
+
1
r
[
∂
S
r
θ
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
r
⊗
e
θ
⊗
e
θ
+
∂
S
r
θ
∂
z
e
r
⊗
e
θ
⊗
e
z
+
∂
S
r
z
∂
r
e
r
⊗
e
z
⊗
e
r
+
1
r
[
∂
S
r
z
∂
θ
−
S
θ
z
]
e
r
⊗
e
z
⊗
e
θ
+
∂
S
r
z
∂
z
e
r
⊗
e
z
⊗
e
z
+
∂
S
θ
r
∂
r
e
θ
⊗
e
r
⊗
e
r
+
1
r
[
∂
S
θ
r
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
θ
⊗
e
r
⊗
e
θ
+
∂
S
θ
r
∂
z
e
θ
⊗
e
r
⊗
e
z
+
∂
S
θ
θ
∂
r
e
θ
⊗
e
θ
⊗
e
r
+
1
r
[
∂
S
θ
θ
∂
θ
+
(
S
r
θ
+
S
θ
r
)
]
e
θ
⊗
e
θ
⊗
e
θ
+
∂
S
θ
θ
∂
z
e
θ
⊗
e
θ
⊗
e
z
+
∂
S
θ
z
∂
r
e
θ
⊗
e
z
⊗
e
r
+
1
r
[
∂
S
θ
z
∂
θ
+
S
r
z
]
e
θ
⊗
e
z
⊗
e
θ
+
∂
S
θ
z
∂
z
e
θ
⊗
e
z
⊗
e
z
+
∂
S
z
r
∂
r
e
z
⊗
e
r
⊗
e
r
+
1
r
[
∂
S
z
r
∂
θ
−
S
z
θ
]
e
z
⊗
e
r
⊗
e
θ
+
∂
S
z
r
∂
z
e
z
⊗
e
r
⊗
e
z
+
∂
S
z
θ
∂
r
e
z
⊗
e
θ
⊗
e
r
+
1
r
[
∂
S
z
θ
∂
θ
+
S
z
r
]
e
z
⊗
e
θ
⊗
e
θ
+
∂
S
z
θ
∂
z
e
z
⊗
e
θ
⊗
e
z
+
∂
S
z
z
∂
r
e
z
⊗
e
z
⊗
e
r
+
1
r
∂
S
z
z
∂
θ
e
z
⊗
e
z
⊗
e
θ
+
∂
S
z
z
∂
z
e
z
⊗
e
z
⊗
e
z
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rr}}{\partial \theta }}-(S_{\theta r}+S_{r\theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rr}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{r\theta }}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rz}}{\partial \theta }}-S_{\theta z}\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta r}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta r}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta \theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta \theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta z}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta z}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{z\theta }}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{z\theta }}{\partial \theta }}+S_{zr}\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zz}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}~{\frac {\partial S_{zz}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}}
Divergence of a second-order tensor field [ edit ] The divergence of a second-order tensor field in cylindrical polar coordinates can be obtained from the expression for the gradient by collecting terms where the scalar product of the two outer vectors in the dyadic products is nonzero. Therefore,
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{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{z}\\[8pt]&+{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\end{aligned}}}
Notes
^ a b c Green, A. E.; Zerna, W. (1968). Theoretical Elasticity . Oxford University Press. ISBN 0-19-853486-8 ^ a b c Ogden, R. W. (2000). Nonlinear elastic deformations . Dover. ^ Naghdi, P. M. (1972). "Theory of shells and plates". In S. Flügge (ed.). Handbook of Physics . Vol. VIa/2. pp. 425–640. ^ a b c d e f g h i j k Simmonds, J. G. (1994). A brief on tensor analysis . Springer. ISBN 0-387-90639-8 ^ a b Basar, Y.; Weichert, D. (2000). Numerical continuum mechanics of solids: fundamental concepts and perspectives . Springer. ^ a b c Ciarlet, P. G. (2000). Theory of Shells . Vol. 1. Elsevier Science. ^ Einstein, A. (1915). "Contribution to the Theory of General Relativity". In Laczos, C. (ed.). The Einstein Decade . p. 213. ^ Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). Gravitation . W. H. Freeman and Co. ISBN 0-7167-0344-0 ^ Greenleaf, A.; Lassas, M.; Uhlmann, G. (2003). "Anisotropic conductivities that cannot be detected by EIT". Physiological Measurement . 24 (2): 413–419. doi :10.1088/0967-3334/24/2/353 . PMID 12812426 . S2CID 250813768 . ^ Leonhardt, U.; Philbin, T. G. (2006). "General relativity in electrical engineering". New Journal of Physics . 8 (10): 247. arXiv :cond-mat/0607418 Bibcode :2006NJPh....8..247L . doi :10.1088/1367-2630/8/10/247 . S2CID 12100599 . ^ "The divergence of a tensor field" . Introduction to Elasticity/Tensors . Wikiversity . Retrieved 2010-11-26 .
Further reading
Spiegel, M. R. (1959). Vector Analysis . New York: Schaum's Outline Series. ISBN 0-07-084378-3 Arfken, George (1995). Mathematical Methods for Physicists . Academic Press. ISBN 0-12-059877-9
Two dimensional Three dimensional
![{\displaystyle {\mathcal {E}}_{ijk}=\left[\mathbf {b} _{i},\mathbf {b} _{j},\mathbf {b} _{k}\right]=(\mathbf {b} _{i}\times \mathbf {b} _{j})\cdot \mathbf {b} _{k}~;~~{\mathcal {E}}^{ijk}=\left[\mathbf {b} ^{i},\mathbf {b} ^{j},\mathbf {b} ^{k}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff8cee95f0d851c6e17855f9efa90baacfcb500)
![{\displaystyle \mathbf {u} \times \mathbf {v} =[(\mathbf {b} _{m}\times \mathbf {b} _{n})\cdot \mathbf {b} _{s}]u^{m}v^{n}\mathbf {b} ^{s}={\mathcal {E}}_{smn}u^{m}v^{n}\mathbf {b} ^{s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbed721be25f93b6f46d2152fcadb59c91738f2)
![{\displaystyle \mathbf {u} \times \mathbf {v} =\varepsilon _{ijk}{\hat {u}}_{j}{\hat {v}}_{k}\mathbf {e} _{i}=\varepsilon _{ijk}{\frac {\partial x_{j}}{\partial q^{m}}}{\frac {\partial x_{k}}{\partial q^{n}}}{\frac {\partial x_{i}}{\partial q^{s}}}u^{m}v^{n}\mathbf {b} ^{s}=[(\mathbf {b} _{m}\times \mathbf {b} _{n})\cdot \mathbf {b} _{s}]u^{m}v^{n}\mathbf {b} ^{s}={\mathcal {E}}_{smn}u^{m}v^{n}\mathbf {b} ^{s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6b6723d4ecd4b178fad61efa6ea6b0e8908c08)
![{\displaystyle \left[{\boldsymbol {S}}\mathbf {u} ,{\boldsymbol {S}}\mathbf {v} ,{\boldsymbol {S}}\mathbf {w} \right]=\det {\boldsymbol {S}}\left[\mathbf {u} ,\mathbf {v} ,\mathbf {w} \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/761265df3679bf22a8fe71a44e5b32b09d00c596)
![{\displaystyle \left[\mathbf {u} ,\mathbf {v} ,\mathbf {w} \right]:=\mathbf {u} \cdot (\mathbf {v} \times \mathbf {w} ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11e2a58fd353028bb79dda5a46e874563e9ab21a)
![{\displaystyle \mathbf {b} ^{i}={\boldsymbol {F}}^{-{\rm {T}}}\mathbf {e} ^{i}~;~~g^{ij}=[{\boldsymbol {F}}^{-{\rm {1}}}{\boldsymbol {F}}^{-{\rm {T}}}]_{ij}~;~~g_{ij}=[g^{ij}]^{-1}=[{\boldsymbol {F}}^{\rm {T}}{\boldsymbol {F}}]_{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b81b661839bdb29f15377c3c046833a29b88ffb)
![{\displaystyle \left[\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right]=\det {\boldsymbol {F}}\left[\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\right]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27fb40b2c4c5aff231420ea55a9fa2087acb5869)
![{\displaystyle \left[\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\right]=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9acb2516827df5a7fa7b889b38161ea0c7edcd29)
![{\displaystyle J=\det {\boldsymbol {F}}=\left[\mathbf {b} _{1},\mathbf {b} _{2},\mathbf {b} _{3}\right]=\mathbf {b} _{1}\cdot (\mathbf {b} _{2}\times \mathbf {b} _{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c246b146b59a610afbd5f727ed1d99ddd6dba4cb)
![{\displaystyle g:=\det[g_{ij}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/862e1ae368f9114d096d6a73ea9b1c0e1dfe3863)
![{\displaystyle g=\det[{\boldsymbol {F}}^{\rm {T}}]\cdot \det[{\boldsymbol {F}}]=J\cdot J=J^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18f8d6db9ee13068f9021b8025f691bfdc225f1e)
![{\displaystyle \det[g^{ij}]={\cfrac {1}{J^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ddad469d755523d50e3eaad1cb5076f61f754e3)
![{\displaystyle [g^{ij}]=[g_{ij}]^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4a027ea5efffa5271d721a669f1ead309ccec8)
![{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})~;~~q^{i}=\psi ^{i}(\mathbf {x} )=[{\boldsymbol {\varphi }}^{-1}(\mathbf {x} )]^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70c34e0633724c76fdaf5108d9c5cefd75527ba5)
![{\displaystyle f(\mathbf {x} )=f[{\boldsymbol {\varphi }}(q^{1},q^{2},q^{3})]=f_{\varphi }(q^{1},q^{2},q^{3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e2cb47b661ab275c69673064f911f3c6952e88)
![{\displaystyle [{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\rm {d}}{\rm {{d}\alpha }}}f(\mathbf {x} +\alpha \mathbf {c} ){\biggr |}_{\alpha =0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32e2cf640dacbf5040ea2d684b6ae6dfe899df71)
![{\displaystyle [{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\rm {d}}{\rm {{d}\alpha }}}f_{\varphi }(q^{1}+\alpha ~c^{1},q^{2}+\alpha ~c^{2},q^{3}+\alpha ~c^{3}){\biggr |}_{\alpha =0}={\cfrac {\partial f_{\varphi }}{\partial q^{i}}}~c^{i}={\cfrac {\partial f}{\partial q^{i}}}~c^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd5094724ed305233cbe03a4e794bc8da9a6971)
![{\displaystyle [{\boldsymbol {\nabla }}\psi ^{i}(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial \psi ^{i}}{\partial q^{j}}}~c^{j}=c^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/847afff4038df4bb39dbd5e678418aff6499c908)
![{\displaystyle [{\boldsymbol {\nabla }}f(\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial f}{\partial q^{i}}}~c^{i}=\left({\cfrac {\partial f}{\partial q^{i}}}~\mathbf {b} ^{i}\right)\left(c^{i}~\mathbf {b} _{i}\right)\quad \Rightarrow \quad {\boldsymbol {\nabla }}f(\mathbf {x} )={\cfrac {\partial f}{\partial q^{i}}}~\mathbf {b} ^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98b14602ba6c3cdd88ec4de14a3a127961863024)
![{\displaystyle [{\boldsymbol {\nabla }}\mathbf {f} (\mathbf {x} )]\cdot \mathbf {c} ={\cfrac {\partial \mathbf {f} }{\partial q^{i}}}~c^{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff66797d897da6fcaf85f3cd5be458b47ad2947)
![{\displaystyle \Gamma _{ijk}={\frac {1}{2}}(g_{ik,j}+g_{jk,i}-g_{ij,k})={\frac {1}{2}}[(\mathbf {b} _{i}\cdot \mathbf {b} _{k})_{,j}+(\mathbf {b} _{j}\cdot \mathbf {b} _{k})_{,i}-(\mathbf {b} _{i}\cdot \mathbf {b} _{j})_{,k}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71d392a16c26aff559865df3e48ad9d7f3038d76)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\mathbf {v} &=\left[{\cfrac {\partial v^{i}}{\partial q^{k}}}+\Gamma _{lk}^{i}~v^{l}\right]~\mathbf {b} _{i}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial v_{i}}{\partial q^{k}}}-\Gamma _{ki}^{l}~v_{l}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} ^{k}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/685b79a086642ff7b74216d9820eeaa5f9c22566)
![{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {S}}={\frac {\partial }{\partial q^{k}}}[S_{ij}~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}]\otimes \mathbf {b} ^{k}=\left[{\frac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} ^{j}\otimes \mathbf {b} ^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eae4c049074ffe07d3de29ab7c23b4751b17b93b)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}{\boldsymbol {S}}&=\left[{\cfrac {\partial S^{ij}}{\partial q^{k}}}+\Gamma _{kl}^{i}~S^{lj}+\Gamma _{kl}^{j}~S^{il}\right]~\mathbf {b} _{i}\otimes \mathbf {b} _{j}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{k}}}+\Gamma _{kl}^{i}~S_{~j}^{l}-\Gamma _{kj}^{l}~S_{~l}^{i}\right]~\mathbf {b} _{i}\otimes \mathbf {b} ^{j}\otimes \mathbf {b} ^{k}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~\mathbf {b} ^{i}\otimes \mathbf {b} _{j}\otimes \mathbf {b} ^{k}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb600be33948beb2496b92771f836436d221eaa2)
![{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\cfrac {\partial v^{i}}{\partial q^{i}}}+\Gamma _{\ell i}^{i}~v^{\ell }=\left[{\cfrac {\partial v_{i}}{\partial q^{j}}}-\Gamma _{ji}^{\ell }~v_{\ell }\right]~g^{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b6988b2d766476c9d8e96fed44b54211daa91cb)
![{\displaystyle \Gamma _{\ell i}^{i}=\Gamma _{i\ell }^{i}={\cfrac {g^{mi}}{2}}\left[{\frac {\partial g_{im}}{\partial q^{\ell }}}+{\frac {\partial g_{\ell m}}{\partial q^{i}}}-{\frac {\partial g_{il}}{\partial q^{m}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b047dc2304c669fa1ac0c7f7f1a764eb7a0e6e)
![{\displaystyle [g_{ij}]^{-1}=[g^{ij}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa63db94365634cc1cdb250153352f3c52fb119)
![{\displaystyle [g^{ij}]=[g_{ij}]^{-1}={\cfrac {A^{ij}}{g}}~;~~g:=\det([g_{ij}])=\det {\boldsymbol {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/421dea6d9ffb9c4b1c8145836b1f6b50f7da5e01)
![{\displaystyle g=\det([g_{ij}])=\sum _{i}g_{ij}~A^{ij}\quad \Rightarrow \quad {\frac {\partial g}{\partial g_{ij}}}=A^{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c507da82e6929c55024bbe9023132dfb42b3f7)
![{\displaystyle [g^{ij}]={\cfrac {1}{g}}~{\frac {\partial g}{\partial g_{ij}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d562b6ccba7d0e712ab74ce7a4dcce58fa1d86a3)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left[{\cfrac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~g^{ik}~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S^{ij}}{\partial q^{i}}}+\Gamma _{il}^{i}~S^{lj}+\Gamma _{il}^{j}~S^{il}\right]~\mathbf {b} _{j}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{i}}}+\Gamma _{il}^{i}~S_{~j}^{l}-\Gamma _{ij}^{l}~S_{~l}^{i}\right]~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~g^{ik}~\mathbf {b} _{j}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85fad1b0228d189867b87f1d770c3968e4394860)
![{\displaystyle \nabla ^{2}\varphi ={\cfrac {1}{\sqrt {g}}}~{\frac {\partial }{\partial q^{i}}}([{\boldsymbol {\nabla }}\varphi ]^{i}~{\sqrt {g}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4050e9f0a19cfbad5ba6dc5cf8a9b255b9ef9b)
![{\displaystyle {\boldsymbol {\nabla }}\varphi ={\frac {\partial \varphi }{\partial q^{l}}}~\mathbf {b} ^{l}=g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}~\mathbf {b} _{i}\quad \Rightarrow \quad [{\boldsymbol {\nabla }}\varphi ]^{i}=g^{li}~{\frac {\partial \varphi }{\partial q^{l}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c37e69ce68c3bb110a443796e6f27fb3e8e4deec)
![{\displaystyle {\begin{aligned}\left|{\partial \mathbf {x} \over \partial t}\right|&=\left|\sum _{k}\left(\sum _{i}h_{ki}~{\cfrac {\partial q^{i}}{\partial t}}\right)\mathbf {e} _{k}\right|\\[8pt]&={\sqrt {\sum _{i}\sum _{j}\sum _{k}h_{ki}~h_{kj}{\cfrac {\partial q^{i}}{\partial t}}{\cfrac {\partial q^{j}}{\partial t}}}}={\sqrt {\sum _{i}\sum _{j}g_{ij}~{\cfrac {\partial q^{i}}{\partial t}}{\cfrac {\partial q^{j}}{\partial t}}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df87fcb92b899e0c92a5406fa23fcf57240f9c77)
![{\displaystyle {\begin{aligned}\left|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right|&=\left|\sum _{k}\sum _{m}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{k}\times \mathbf {e} _{m}\right|\\[8pt]&=\left|\sum _{p}\sum _{k}\sum _{m}{\mathcal {E}}_{kmp}\left(\sum _{i=1}^{3}h_{ki}~{\partial q^{i} \over \partial s}\right)\left(\sum _{j=1}^{3}h_{mj}~{\partial q^{j} \over \partial t}\right)\mathbf {e} _{p}\right|\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f57e1fe69fd45a46a61b013df668d3921ffcb09)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{r}}{\partial \theta }}-v_{\theta }\right)~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{r}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\cfrac {\partial v_{\theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left({\cfrac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{\theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\cfrac {\partial v_{z}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}{\cfrac {\partial v_{z}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\cfrac {\partial v_{z}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d7285d16335aefea8e8f9682612e397efb70663)
![{\displaystyle \mathbf {v} ={\boldsymbol {\nabla }}f=\left[v_{r}~~v_{\theta }~~v_{z}\right]=\left[{\cfrac {\partial f}{\partial r}}~~{\cfrac {1}{r}}{\cfrac {\partial f}{\partial \theta }}~~{\cfrac {\partial f}{\partial z}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b40b85224ce20bf037fac2e02a46395ee0ad75f)
![{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\boldsymbol {\nabla }}^{2}f={\cfrac {\partial ^{2}f}{\partial r^{2}}}+{\cfrac {1}{r}}\left({\cfrac {1}{r}}{\cfrac {\partial ^{2}f}{\partial \theta ^{2}}}+{\cfrac {\partial f}{\partial r}}\right)+{\cfrac {\partial ^{2}f}{\partial z^{2}}}={\cfrac {1}{r}}\left[{\cfrac {\partial }{\partial r}}\left(r{\cfrac {\partial f}{\partial r}}\right)\right]+{\cfrac {1}{r^{2}}}{\cfrac {\partial ^{2}f}{\partial \theta ^{2}}}+{\cfrac {\partial ^{2}f}{\partial z^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4140151e9340dc7f835e7cb6b745ddc3f0c70ef3)
![{\displaystyle {\begin{aligned}{\hat {S}}_{11}&=S_{11}=:S_{rr},&{\hat {S}}_{12}&={\frac {S_{12}}{r}}=:S_{r\theta },&{\hat {S}}_{13}&=S_{13}=:S_{rz}\\[6pt]{\hat {S}}_{21}&={\frac {S_{21}}{r}}=:S_{\theta r},&{\hat {S}}_{22}&={\frac {S_{22}}{r^{2}}}=:S_{\theta \theta },&{\hat {S}}_{23}&={\frac {S_{23}}{r}}=:S_{\theta z}\\[6pt]{\hat {S}}_{31}&=S_{31}=:S_{zr},&{\hat {S}}_{32}&={\frac {S_{32}}{r}}=:S_{z\theta },&{\hat {S}}_{33}&=S_{33}=:S_{zz}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df61068b926a6e824c7ab7bd30c7f5e63e6d70ee)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rr}}{\partial \theta }}-(S_{\theta r}+S_{r\theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rr}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{r\theta }}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{rz}}{\partial \theta }}-S_{\theta z}\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta r}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta r}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta \theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta \theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{\theta z}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{\theta z}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{z\theta }}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{z\theta }}{\partial \theta }}+S_{zr}\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zz}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\cfrac {1}{r}}~{\frac {\partial S_{zz}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e4bfc2558587fc7fc8f8380c1fdbf7c7310ee6)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{z}\\[8pt]&+{\cfrac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }+{\cfrac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{z}\\[8pt]&+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f63523689efdcb26ba570073d9620fc0da086a0a)