Summary of the Calculus of Vector Fields

In a Nut Shell: There are several helpful theorems in multi-variable calculus of

vector fields. They enable you to pick alternative ways to evaluate line integrals and

surface integrals. So doing you can select the easiest option for your problem.

A Line Integral, I_L, is used to evaluate the value of a vector function, F(x,y), along a

plane curve, C, or of a vector function, F(x,y,z), along a space curve, C.

I_L = ∫ F . dr = ∫ F . (dr/dt) dt

C C

The Surface Integral, I_s, is analogous to the line integral in that it provides the value of

a function, f(x,y,z), evaluated over a “smooth” surface, S, in space. Here the surface

integral is

I_s = ∫ ∫ f(x, y, z) dS

S

where dS is the element of surface area on the spatial surface.

Green’s Theorem gives the relationship between a line integral around a simple closed

curve in a plane, C, and a double integral over the enclosed plane region R bounded by C.

∫ P dx + Q dy = ∫ ∫ [∂Q/∂x - ∂P/∂y] dA (standard form of Green’s Theorem)

C R

The curve, C, is said to be positively oriented when traveling counterclockwise around

C keeping the region, R, enclosed to the left. If F(x,y) = P(x,y) i + Q(x,y) j and

dr = dx i + dy j, then F . dr = Pdx + Qdy

So ∫ F . dr = ∫ ∫ [∂Q/∂x - ∂P/∂y] dA

C R

Also ∫ F . dr = ∫ F . Tds = ∫ ∫ curl_z F dA (curl form of Green’s Theorem)

C C R

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

C R

Click here to continue with this summary of vector fields.

Return to Calculus 3 Notes

Copyright © 2017 Richard C. Coddington

All rights reserved.