Wikipedia:Reference desk/Archives/Mathematics/2011 January 14

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January 14

Gradients

What does it mean to take a gradient with respect to a bilinear form? If ψ is a smooth function and h is a non-degenerate bilinear form then grad_h(ψ) denotes "the gradient with respect to the non-degenerate bilinear form h". — Fly by Night (talk) 11:48, 14 January 2011 (UTC)

$\left({\frac {\frac {d\psi }{dx}}{\frac {dh}{dx}}},{\frac {\frac {d\psi }{dy}}{\frac {dh}{dy}}}\right)$ I suppose. I am not sure. Bo Jacoby (talk) 13:19, 14 January 2011 (UTC).

That would seem to transform very irregularly under (even orthogonal) basis changes. And what would dh/dx mean when h is a bilinear form? –Henning Makholm (talk) 13:39, 14 January 2011 (UTC)

You are correct. Sorry. I thought of a quadratic form. Bo Jacoby (talk) 14:03, 14 January 2011 (UTC).

The gradient of ψ at p (a point of the domain of ψ, say an open set of the space V) with respect to the non-degenerate bilinear form h (defined on the space V) is just the element of V that corresponds to the linear form Dψ(p) (the Fréchet differential) in the isomorphism (described in the linked article) $V\to V^{*}$ induced by h. Thus, for any x in V we have

h( x, grad_h(ψ))=Dψ(p)[x],

and for instance if h was the standard scalar product of $\mathbb {R} ^{n}$ the gradient is the usual vector of the partial derivatives of ψ. Warning: in your case one may use $v\to h(v,\cdot )$ instead than $v\to h(\cdot ,v)$ to define the isomorphism, which would affect the definition of grad_h(ψ) if h is not symmetric. Also note that this definition of "gradient" is used in infinite dimension too (for instance, the gradient of a functional defined on $L^{p}$ is an element of the conjugated space $L^{q}$; the gradient of a functional on C(K) is a Radon measure on K etc).--pma 14:49, 14 January 2011 (UTC)

Thanks for the reply pma. I'm not sure I follow you, so let's try and unravel it. In my case ψ is a smooth function on a smooth hypersurface M ⊂ Rⁿ. The bilinear form is non-degenerate by assumption (we impose an open and dense condition on the manifold M), and it actually is symmetric, so h(X, Y) = h(Y, X) for all vector fields X and Y on M. In the notation I'm used to, D usually means the covariant derivative on Rⁿ. When you write Dψ(p)[x], is that the same as d_Xψ evaluated at p ∈ M, i.e. the exterior derivative of ψ at p, evaluated on X ∈ T_pM ? If all of this is correct, then what is grad_h(ψ) explicitly? (As a concrete example, consider a smooth surface in R³. One choice for h would be the second fundamental form, (II). The non-degeneracy of h means that we exclude parabolic points. In this case, what is grad_(II)(ψ) for some smooth ψ : M → R. — Fly by Night (talk) 00:35, 15 January 2011 (UTC)

Yes, the notations vary in the various contexts. In any case, the differential of $\psi$ at $p\in M$ ($d\psi (p)$ or $d\psi _{p}$ or $T_{p}\psi$) is a linear form on $T_{p}M$ and it is a purely differential object, in that it is well-defined for any differentiable function on a differentiable manifold. If there is a Riemann structure $g$ on $M$, the natural way to represent the differential at p is using the scalar product $g_{p}$ with a vector (the gradient), as said. The same can be done done with semi-riemannian structures. But, honestly I do not have an idea of what is the meaning of using the second fundamental form to represent a differential of a function, and what is the meaning of the corresponding "II-gradient" vector. --pma 18:01, 15 January 2011 (UTC)

For what it's worth, what I get from pma's explanation, paraphrased into the physicist's notation I have learned (which may or may not actually be helpful to Fly by Night) is this:

The quadratic form $h$ can be represented by a symmetric rank 2 covariant tensor $h_{ij}$. Because it is non-degenerate, it has an inverse which we can call $H^{ij}$, with the property $H^{ij}h_{jk}=\delta _{k}^{i}$. Then the $\mathrm {grad} _{h}(\psi )$ is simply the contravariant vector that results from contracting the ordinary (covariant) gradient of $\psi$ with the inverse of h: $(\mathrm {grad} _{h}(\psi ))^{i}=H^{ij}\psi _{,j}$. –Henning Makholm (talk) 20:39, 15 January 2011 (UTC)

I think you might be right. Our article on the Laplace–Beltrami operatorsays, and I quote... The gradient of a scalar function ƒ is the vector field grad f that may be defined through the inner product $\langle \cdot ,\cdot \rangle$ on the manifold, as

$\langle {\mbox{grad}}f(x),v_{x}\rangle =df(x)(v_{x})$

for all vectors v_x anchored at point x in the tangent bundle T_xM [sic] of the manifold at point x. Here, dƒ is the exterior derivative of the function ƒ; it is a 1-form taking argument v_x. In local coordinates, one has

$\left({\mbox{grad}}f\right)^{i}=\partial ^{i}f=g^{ij}\partial _{j}f\,.$

As pma said: $h({\mbox{grad}}_{h}(\psi ),v)=D\psi (p)[v]$; so we would get

$\left[{\mbox{grad}}_{h}(\psi )\right]^{i}=h^{ij}\partial _{j}\psi \,.$

Nice work chaps. Thank you both for your help. It does turn up quite a lot in different calculations; for example if you have non-standard connections. Now I just need to work out if it has any real meaning or is just used to simplify notation. — Fly by Night (talk) 02:51, 16 January 2011 (UTC)

A notational caveat, possibly with real consequences: In a (pseudo-)Riemannian space the metric tensor g (that is, the first fundamental form if the space is embedded in an Euclidean one) is specially distinguished, and I would expect $h^{ij}$ to mean $h_{ij}$ with both indices raised by the metric, i.e. $g^{ia}h_{ab}g^{bj}$, which is quite a different creature than the inverse of h. This is why I used a capital H for the inverse. –Henning Makholm (talk) 04:05, 16 January 2011 (UTC)

Interesting. I'm sure I've seen $(h^{ij})$ for the inverse of $(h_{ij})$ is differential geometry books. I haven't read any physics books since high school. Although, maybe I had misunderstood what $(h^{ij})$ was used to mean. Anyway, I'll be writing it in a coordinate-free way anyway, so I shouldn't need any subscripts or superscripts. Everything's written using connections, e.g.

$D_{X}Y=\nabla _{X}Y+h(X,Y)\xi \,,$

where D is the covariant derivative of Rⁿ, X and Y are vector fields on a hypersurface M ⊂ Rⁿ, ξ is any transverse vector field on M (an example would be the unit normal), ∇ is an induced (torsion-free) connection of M dependant on the choice of ξ, and h is symmetric bilinear form dependant on the choice of ξ but whose non-degeneracy does not depend upon the choice of ξ. — Fly by Night (talk) 14:10, 16 January 2011 (UTC)

I must confess that I've never managed to become comfortable with that sort of notation. Plain abstract index notation is quite coordinate-free as far as I'm concerned, and so much easier to grasp. In the mathematician's notation I'm never quite sure even which rank/type each expression is supposed to have, without using half an hour crossreferencing all of the symbols and figuring out which of their overloads the author meant in this case. With index notation it's just a matter of counting the unpaired indices: there's the rank plain as day. –Henning Makholm (talk) 00:31, 17 January 2011 (UTC)