DIVERGENCE THEOREM AND STOKES’ THEOREM

University: 香港中文大學

Course: Advanced Calculus II (MATH2020)

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68 Theory Supplement Section M

MPROOF OF THE DIVERGENCE THEOREM AND STOKES’ THEOREM

In this section we give proofs of the Divergence Theorem and Stokes’ Theorem using the definitions

in Cartesian coordinates.

Proof of the Divergence Theorem

Let ~

Fbe a smooth vector field defined on a solid region Vwith boundary surface Aoriented

outward. We wish to show that

ZA

~

F·d~

A=ZV

div ~

F dV.

For the Divergence Theorem, we use the same approach as we used for Green’s Theorem; first

prove the theorem for rectangular regions, then use the change of variables formula to prove it for

regions parameterized by rectangular regions, and finally paste such regions together to form general

regions.

Proof for Rectangular Solids with Sides Parallel to the Axes

Consider a smooth vector field ~

Fdefined on the rectangular solid V:a≤x≤b,c≤y≤d,

e≤z≤f. (See Figure M.50). We start by computing the flux of ~

Fthrough the two faces of V

perpendicular to the x-axis, A1and A2, both oriented outward:

ZA1

~

F·d~

A+ZA2

~

F·d~

A=−Zf

eZd

c

F1(a, y, z)dy dz +Zf

eZd

c

F1(b, y, z)dy dz

=Zf

eZd

c

(F1(b, y, z)−F1(a, y, z)) dy dz.

By the Fundamental Theorem of Calculus,

F1(b, y, z)−F1(a, y, z) = Zb

a

∂F1

∂x dx,

so

ZA1

~

F·d~

A+ZA2

~

F·d~

A=Zf

eZd

cZb

a

∂F1

∂x dx dy dz =ZV

∂F1

∂x dV.

By a similar argument, we can show

ZA3

~

F·d~

A+ZA4

~

F·d~

A=ZV

∂F2

∂y dV and ZA5

~

F·d~

A+ZA6

~

F·d~

A=ZV

∂F3

∂z dV.

Adding these, we get

ZA

~

F·d~

A=ZV∂F1

∂x +∂F2

∂y +∂F3

∂z dV =ZV

div ~

F dV.

This is the Divergence Theorem for the region V.