Summary of the Calculus of Vector Fields (continued)

The Divergence Theorem (Gauss's Theorem) extends the divergence form of Green’s
Theorem from two to three dimensions. In this case the line integral around a closed curve
is replaced by a surface integral around a closed surface and the area integral involving

the divergence of F is replaced by the volume integral of the divergence of F .

2-D

∫ F . n ds = ∫ ∫ div F dA (Divergence form of Green’s Theorem)

C R

3-D

∫ ∫ F . n dS = ∫ ∫ ∫ div F dV (Divergence Theorem)

S E

Also ∫ ∫ F . n dS = ∫ ∫ F . (r_u x r_v) dA (Conversion to a surface integral)

S R

Green's theorem gives the relationship between a line integral around a simple closed

curve, C, in the x-y plane. Stokes’ Theorem extends this concept to a closed curve in

x-y-z space using the “curl form” of Green’s Theorem as follows:

2-D

∫ F (x,y) . dr = ∫ ∫ curl_z F dA = ∫ ∫ curl F . k dA (curl form of Green’s Theorem)

C R R

3-D Green's Theorem extended to Stokes’ Theorem gives :

∫ F (x,y,z) . dr = ∫ ∫ curl F . n dS (Stokes’ Theorem)

C S

where F(x,y,z) = P(x,y,z) i + Q(x,y,z) j + R(x,y,z) k

Also ∫ ∫ curl F . n dS = ∫ ∫ curl F . (r_u x r_v) dA (Conversion to a surface integral)

S R

Return to Calculus 3 Notes

Copyright © 2017 Richard C. Coddington

All rights reserved.