Chapter 4: Partial Differentiation
Section 4.6: Surface Normal and Tangent Plane
Derive the form of N for the surface given implicitly by fx,y,z=c, where c is a real constant.
(See Table 4.6.1.)
The equation fx,y,z=c can, in principle, be solved explicitly for z=zx,y wherever fz≠0.
Implicit (partial) differentiation leads to zx=−fx/fz and zy=−fy/fz.
The coordinate curves R1=xbz(x,b) and R2=ayz(a,y) project onto the grid lines y=b and x=a, respectively.
Tangents to these curves, namely, T1=10zx and T2=01zy, are distinct vectors tangent to the surface.
Their cross product
T1×T2=ijk10zx01zy = ijk10−fx/fz01−fy/fz = fx/fzfy/fz1 = 1fz fxfyfz
is then orthogonal to the surface at the point a,b,c. Since N= ∇f in Table 4.6.1 is a multiple of this vector, it follows that N itself is orthogonal to the surface.
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