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Physics / Introduction to Electrodynamics 4 / Chapter 1 / Problem 60P

Introduction to Electrodynamics | 4th Edition | ISBN: 9780321856562 | Authors: David J. Griffiths

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Here are two cute checks of the fundamental theorems:(a)

Chapter 1, Problem 60P

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QUESTION:

Here are two cute checks of the fundamental theorems:

(a) Combine Corollary 2 to the gradient theorem with Stokes’ theorem ( $v=$ $T$ , in this case). Show that the result is consistent with what you already knew about second derivatives.

(b) Combine Corollary 2 to Stokes’ theorem with the divergence theorem. Show that the result is consistent with what you already knew.

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QUESTION:

Here are two cute checks of the fundamental theorems:

(a) Combine Corollary 2 to the gradient theorem with Stokes’ theorem ( $v=$ $T$ , in this case). Show that the result is consistent with what you already knew about second derivatives.

(b) Combine Corollary 2 to Stokes’ theorem with the divergence theorem. Show that the result is consistent with what you already knew.

ANSWER:

Step 1 of 3

a) Here we need to combine corollary 2 with the gradient theorem to the Stoke’s theorem.

Corollary 2 is: $\oint\limits_{\ }^{\ }($ $\times{}v).da=0$ for any closed surface.

Stokes’ theorem says that,

$\int\limits_{S}^{\ }($ $\times{}v).da=\oint\limits_{\ }^{\ }v.dl$

And finally the gradient theorem is,

$\int\limits_{a}^{b}($ $T)\ .dl=T(b)-T(a)$

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