Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
Prove Property 2 in Table 4.5.1.
Property 2: The gradients of fx,y,z are orthogonal to the level surfaces z=zx,y defined implicitly by fx,y,z=c, where c is a real constant.
Let S be the (level) surface z=zx,y defined implicitly by fx,y,z=c.
Let P:α,β,γ be a point on on S so that fα,β,γ≡c.
Let C1=αyz(α,y) describe the coordinate curve in S that projects onto the grid line x=α.
Let C2=xβz(x,β) describe the coordinate curve in S that projects onto the grid line y=β.
Then T1=01zy and T2=10zx are tangent respectively to C1 and C2.
The gradient ∇f will be orthogonal to S if it is orthogonal to both T1 and T2.
Implicitly differentiate fx,y,zx,y≡c to get fx+fz zx=0 and fy+fz zy=0, from which zx=−fx/fz and zy=−fy/fz then follow.
Thus, T1 and T2 become respectively 01−fy/fz and 10−fx/fz, so that
∇f·T1=fxfyfz·01−fy/fz = fy−fy=0 and ∇f·T2=fxfyfz·10−fx/fz = fx−fx=0
Hence, ∇f is orthogonal to two independent tangent vectors on S, so it is orthogonal to S itself.
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