Kenneth Kuttler

390 CHAPTER 21. THE INTEGRAL ON TWO DIMENSIONAL SURFACES IN R3

Suppose then that x= f (u) where u ∈ U , a subset of R2 and x is a point in V , asubset of 3 dimensional space. Thus, letting the Cartesian coordinates of x be given byx = (x1,x2,x3)

T , each xi being a function of u, an infinitesimal rectangle located at u0corresponds to an infinitesimal parallelogram located at f (u0) which is determined by the

2 vectors{

∂f(u0)∂ui

dui

i=1, each of which is tangent to the surface defined by x= f (u).

(No sum on the repeated index.)

du2

du1

fu2(u0)du2

fu1(u0)du1

f(dV )

From Definition 21.1.1, the two dimensional volume of this infinitesimal parallelepipedlocated at f (u0) is given by∣∣∣∣∂f (u0)

∂u1du1×

∂f (u0)

∂u2du2

∣∣∣∣ =

∣∣∣∣∂f (u0)

∂u1× ∂f (u0)

∂u2

∣∣∣∣du1du2 (21.1)

=∣∣fu1×fu2

∣∣du1du2 (21.2)

It might help to think of a lizard. The infinitesimal parallelepiped is like a very smallscale on a lizard. This is the essence of the idea. To define the area of the lizard sum upareas of individual scales1. If the scales are small enough, their sum would serve as a goodapproximation to the area of the lizard.

This motivates the following fundamental procedure which I hope is extremely familiarfrom the earlier material.

1This beautiful lizard is a Sceloporus magister. It was photographed by C. Riley Nelson who is in the Zoologydepartment at Brigham Young University © 2004 in Kane Co. Utah. The lizard is a little less than one foot inlength.

390 CHAPTER 21. THE INTEGRAL ON TWO DIMENSIONAL SURFACES IN R°*Suppose then that 2 = f (uw) where u € U, a subset of R? and 2 is a point in V, asubset of 3 dimensional space. Thus, letting the Cartesian coordinates of a be given byx= (x ,X2,%3), each x; being a function of w, an infinitesimal rectangle located at upcorresponds to an infinitesimal parallelogram located at f (uo) which is determined by the22 vectors { 2f{u0) dui}U i=(No sum on the repeated index.), each of which is tangent to the surface defined by x = f (u).f(dv)dv Puy (tbody aUo du, aoe .., aduyFrom Definition 21.1.1, the two dimensional volume of this infinitesimal parallelepipedlocated at f (uo) is given byof tuo) duj x of tuo) au = eae x oF (uo) duyduy (21.1)1 2 1 2fu x Fu. |duiduy (21.2)It might help to think of a lizard. The infinitesimal parallelepiped is like a very smallscale on a lizard. This is the essence of the idea. To define the area of the lizard sum upareas of individual scales! . If the scales are small enough, their sum would serve as a goodapproximation to the area of the lizard.This motivates the following fundamental procedure which I hope is extremely familiarfrom the earlier material.'This beautiful lizard is a Sceloporus magister. It was photographed by C. Riley Nelson who is in the Zoologydepartment at Brigham Young University © 2004 in Kane Co. Utah. The lizard is a little less than one foot inlength.