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March 14

Jacobians and vector field differentials

Am I correct in saying that the Jacobian matrix is essentially taking the gradient of each component of a vector field?

In what sense does $d\mathbf {w} =J_{\mathbf {w} }\cdot d\mathbf {x}$ ? There, w is a vector field, x is position (in column vector form, for matrix multiplication purposes), and the dot product not a real dot product but an "inner product" (I'm not too sure of this) in the sense that each entry of the Jacobian appears represent the (scalar) product of a partial derivative and a dyadic unit tensor for each unit vector pair (according to an expert I asked). Further, am I correct in asserting that taking that "inner product" of the Jacobian with a vector produces a directional derivative of the vector field?

The dyadic tensor article is of little help for me because "juxtaposition" (of vectors) is not something that makes sense to me.--Jasper Deng (talk) 05:28, 14 March 2014 (UTC)[reply]

If you take "vector field" to mean function on Rⁿ with values in R^m, where each of the domain and codomain are represented as column vectors, then the Jacobian matrix of such a function

\mathbf {w} :\mathbb {R} ^{n}\to \mathbb {R} ^{m}

is the m×n matrix whose rows are the gradients of the components of w. In the formula

d\mathbf {w} =J_{\mathbf {w} }\cdot d\mathbf {x}

, the "dot" is ordinary matrix multiplication of an m×n matrix by an n×1 column vector. (This is actually a form of the chain rule.) Finally, if u is a unit vector in Rⁿ, then the directional derivative of w along u is

J_{\mathbf {w} }\mathbf {u}

.

All of this assumes that you are working in Euclidean space with a Cartesian coordinate system. It is no longer true in a general coordinate system (or on a manifold), since strictly speaking the gradient doesn't transform correctly under coordinate transitions. It is contravariant, which means that it transforms as a column vector rather than a row vector. Instead of the gradient, you need to use the differential, which transforms as a row vector. Sławomir Biały (talk) 11:52, 16 March 2014 (UTC)[reply]

how to replace 1.66^-18 to scientific form?

I know that it equals to 1.66*10^-17 (because someone tell me that...), but how can I understand and do that by myself? May I do that by my calculator? (I have fx82 of Casio) 5.28.179.184 (talk) 23:40, 14 March 2014 (UTC)[reply]

First of all, if you mean

1.66^{-18}

, then that is not even correct - that equals about 0.00010915771. --Jasper Deng (talk) 04:52, 15 March 2014 (UTC)[reply]

Please double-check the problem; if it actually states

1.66\times 10^{-18}

, then that is already in proper scientific notation. --Kinu ^t/_c 04:59, 15 March 2014 (UTC)[reply]

Sorry, I don't understand what the problem is. The origin question is "How many moles - million molecules water they are". So, the formula is:

n={\frac {N}{Avogadro}}

, then, according this formula we get:

1.66^{-18}={\frac {10^{6}}{6.02*10^{23}}}

. Someone tell me (and I saw that he's right) that

1.66^{-18}=1/6*10^{-17}

. My question was (and is) how can I reach to the scientific form (

1/6*10^{-17}

) by the calculator, not by someone else...5.28.179.184 (talk) 08:38, 15 March 2014 (UTC)[reply]

The answer is

1.66\times 10^{-18}

(to 3 significant figures). This is already in scientific notation form. The expression

1.66^{-18}

, although it looks similar, has a very different value. What may be confusing you is that your calculator probably displays the answer as

1.66{\text{E}}{-18}

, which is

1.66\times 10^{-18}

in E notation. Gandalf61 (talk) 10:56, 15 March 2014 (UTC)[reply]

Actually, the calculator likely does display it as

1.66^{-18}

(that's how it looks on my Casio calculator). That's pretty harmless if you know what you're doing and how the calculator displays information, but it can certainly be a source of confusion. -- Meni Rosenfeld (talk) 17:30, 16 March 2014 (UTC)[reply]

Are you sure there isn't a tiny ×10 between 1.66 and -18? The online Casio calculator documentation I found shows that, including for Casio fx82MS. PrimeHunter (talk) 18:43, 16 March 2014 (UTC)[reply]

My Casio fx-7400G Plus displays scientific numbers without the ×10 or an E.--Salix alba (talk): 18:56, 16 March 2014 (UTC)[reply]

OK, it appears to vary. See [1] for a tiny ×10 in a Casio fx-300MS. PrimeHunter (talk) 19:12, 16 March 2014 (UTC)[reply]

Some calculators do not display the x10 see these pictures.

https://lh5.ggpht.com/QP2gGoYmn9hOjFqc1Hng4hI7mNjQoURc6YrVbgTXpluKBTnzhoFiS25I-YJJHW5Afw=h900

https://lh4.ggpht.com/RNUDeFEdrZhzQ3L-EJvULhx2Y2mO7iuDdNJHjfuNrpcsblic6mP5IU3kWbPz0LVYuU8=h900

202.177.218.59 (talk) 22:52, 16 March 2014 (UTC)[reply]