Chapter 4: Partial Differentiation
Section 4.6: Surface Normal and Tangent Plane
Derive the form of N for the surface given explicitly by z=fx,y. (See Table 4.6.1.)
The coordinate curves R1=xbf(x,b) and R2=ayf(a,y) project onto the grid lines y=b and x=a, respectively.
Tangents to these curves, namely, T1=10fx and T2=01fy, are distinct vectors tangent to the surface.
Their cross product
T1×T2=ijk10fx01fy = −fx−fy1
is then orthogonal to the surface at the point a,b,fa,b, that is, it is the normal N listed in Table 4.6.1.
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