Kenneth Kuttler

558 CHAPTER 28. THE INTEGRAL ON SURFACES

du2

du1

fu2(u0)du2

fu1(u0)du1

f(dV )

From Definition 28.1.1, 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 (28.1)

=∣∣fu1

×fu2

∣∣du1du2 (28.2)

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.

Procedure 28.1.2 Suppose U is a subset of R2 and suppose f : U → f (U) ⊆ R3

is a one to one and C1 function. Then if h : f (U)→ R, define the 2 dimensional surfaceintegral

∫f(U) h(x) dA according to the following formula.∫

f(U)h(x) dA ≡

∫U

h(f (u))∣∣fu1

(u)×fu2(u)∣∣du1du2

=∫

Uh(f (u))det(G(u))1/2 du1du2

where G(u) =

(fu1

·fu1fu1

·fu2fu1

·fu2fu2

·fu2

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.

558 CHAPTER 28. THE INTEGRAL ON SURFACESduydVuo du,From Definition 28.1.1, the two dimensional volume of this infinitesimal parallelepipedlocated at f (ug) is given byoF (uo) du, x oF (wo) au = oe x oF (wo) du; du (28.1)1 2 1 2lf X Fun | duidur (28.2)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.Procedure 28.1.2 suppose U is a subset of R2 and suppose f :U > f (U) CRis a one to one and C! function. Then if h: f (U) > R, define the 2 dimensional surfaceintegral Jew) h(a) dA according to the following formula.[ut dA| [ (F(t) [Fy (tH) X Fug (tt)| derrdup[WF (a) det(G (u))"? daarwhere G(u) = ( fa Je fa Je ) .'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.