Chapter 6: Applications of Double Integration
Section 6.3: Surface Area
Derive the expression for dσ when the surface is given implicitly. See Table 6.3.2.
In principle, the surface zx,y defined implicitly by the equation hx,y,z=0 can be obtained explicitly by solving h=0 for z=fx,y, in which case hx,y,fx,y≡0, that is, the equation becomes an identity in x and y.
Implicit differentiation of this identity leads to
hx+hz fx=0⇒fx= −hxhz and hy+hz fy=0⇒fy= −hyhz
Use of λ=1+fx2+fy2, valid for the explicitly given surface z=fx,y, leads to
λ=1+−hxhz2+−hyhz2 = hx+hy2+hz2hz2 = hx+hy2+hz2hz = ∇h|hz|
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