Mathematical Tools for Physics - Department of Physics - University

Vector Calculus 2

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There’s more to the subject of vector calculus than the material in chapter nine. There are a couple
of types of line integrals and there are some basic theorems that relate the integrals to the derivatives,
sort of like the fundamental theorem of calculus that relates the integral to the anti-derivative in one
dimension.

13.1 Integrals
Recall the definition of the Riemann integral from section1.6.

∫b

a

dxf(x) = lim

∆xk→ 0

∑N

k=1

f(ξk) ∆xk (13.1)

This refers to a function of a single variable, integrated along that one dimension.
The basic idea is that you divide a complicated thing into little pieces to get an approximate
answer. Then you refine the pieces into still smaller ones to improve the answer and finally take the
limit as the approximation becomes perfect.
What is the length of a curve in the plane? Divide the curve into a lot of small pieces, then if the
pieces are small enough you can use the Pythagorean Theorem to estimate the length of each piece.

∆`k=

√

(∆xk)^2 + (∆yk)^2

∆xk

∆yk

The whole curve then has a length that you estimate to be the sum of all these intervals. Finally take
the limit to get the exact answer.

∑

k

∆`k=

∑√

(∆xk)^2 + (∆yk)^2 −→

∫

d`=

∫ √

dx^2 +dy^2 (13.2)

How do you actuallydothis? That will depend on the way that you use to describe the curve itself.
Start with the simplest method and assume that you have a parametric representation of the curve:

x=f(t) and y=g(t)

Thendx=f ̇(t)dtanddy=g ̇(t)dt, so

d`=

√(

f ̇(t)dt)^2 +(g ̇(t)dt)^2 =

√

f ̇(t)^2 +g ̇(t)^2 dt (13.3)

and the integral for the length is

∫

d`=

∫b

a

dt

√

f ̇(t)^2 +g ̇(t)^2

whereaandbare the limits on the parametert. Think of this as

∫

d`=

∫

vdt, wherevis the speed.

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