Mathematical Tools for Physics

13—Vector Calculus 2 402

Now divide this volume into a lot of little volumes,∆Vk with individual bounding surfacesSk. If you do the
surface integrals of~v.dA~over each of these pieces and add all of them, the result is the original surface integral.

∑

k

∮

Sk

~v.dA~=

∮

S

~v.dA~ (12)

The reason for this is that each interior face of volumeVkis matched with the face of an adjoining volumeVk′.
The latter face will havedA~pointing in the opposite direction, so when you add all the interior surface integrals
they cancel. All that’s left is the surface on the outside and the sum over allthosefaces is the original surface
integral.
In the equation ( 12 ) multiply and divide every term in the sum by the volume∆Vk.

∑

k

[

1

∆Vk

∮

Sk

~v.dA~

]

∆Vk=

∮

S

~v.dA~

Now increase the number of subdivisions, finally taking the limit as all the∆Vkapproach zero. The quantity
inside the brackets becomes the definition of the divergence of~vand you then get

Gauss’s Theorem:

∫

V

div~v dV =

∮

S

~v.dA~ (13)

This* is Gauss’s theorem, the divergence theorem.

13.4 Stokes’ Theorem
The expression for the curl in terms of integrals is Eq. (9.17),

curl~v= lim

V→ 0

1

V

∮

dA~×~v (14)

• You will sometimes see the notation∂V instead ofSfor the boundary surface surrounding the volumeV.
  Also∂Ainstead ofCfor the boundary curve surrounding the areaA. It’s probably a better and more consistent
  notation, but it isn’t yet as common in physics books.