5c-lecture-07-19-17

Surface Area

S is a surface

z=f(x y)
grad(f) exists

Divide D into small rectangles Rij with area dA
Pij(xi,yi,f(xi,yi)) is a point in S. 

The tangent plane Tij to S approximates S near Pij. dTij approximates dSij at Rij.

a and b are the vectors that start at Pij and lie along the sides of the parallelogram with area dTij.

Thus SSdTij approximates the total area of S. 

And the Surface Area of S is A(S) = lim m,n-> infinity SS dTij observe the area of the parallelogram is |axb|.

Pij(xi,yj,f(xi,yj))

z-f(xi yj)=fx(xi yj)(x-xi)+fy(xi yj)(y-yj)
L(xi+dx yj)= f(xi yj)(xi -xi)+ fx(xi yj)(xi +dx -xi)+  fy(xi yj)(yj-yj)
L(xi+dx yj)=f(xi yj)+fx(xi yi)dx

=> a=<dx, 0,fx(xi yj)dx> b=<0,dy,fx(xi yi)dy>
a=<dx, 0,fx(xi yj)dx> 
b=<0,dy,fx(xi yi)dy>

axb=<-fx(xi yj)dxdy, -fy(xi yj)dxdy, dxdy>
|axb|=sqrt((fx(xi yj))^2+fy(xi yj)^2+1)dxdy

=> dS=sqrt((fx(xi yj))^2+fy(xi yj)^2+1)dxdy

A(S)= SS(dS)