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Continue reading...Introduction

This article will attempt to take the mystery out of setting up surface integrals. It will explain the basic ideas underlying surface integration and show you how to parameterize surfaces to set up the corresponding integrals efficiently. You will need to have had or be taking ##3D## calculus covering multiple integrals to get the most out this presentation. The focus is on how to set up the integrals but not so much on techniques of evaluating them.

Vector functions giving space curves

Readers will be familiar with the vector function $$\vec R(t)= \langle x(t),y(t) \rangle = \langle 2\cos t, 2\sin t\rangle,~0\le t \le 2\pi$$which maps the ##t## interval ##I=[0,2\pi]## onto a circle of radius ##2## in the ##x,y## plane. Here, the value of the single independent variable ##t## determines the location of a point on the circle. I will call the interval ##I## the “parameter space”.

Similarly, a vector function of the form$$\vec R(t) = \langle x(t),y(t),z(t)\rangle,~a \le t...

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