Mathematical Tools for Physics

Vector Calculus 2

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 (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.

k

D

D

x

y

k

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.

∆sk=

√

(∆xk)^2 + (∆yk)^2

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

∆sk=

∑√

(∆xk)^2 + (∆yk)^2 −→

∫

ds=

∫ √

dx^2 +dy^2 (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)

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