Chapter 4: Partial Differentiation
Section 4.5: Gradient Vector
Prove Property 1 in Table 4.5.1.
Property 1: The gradients of fx,y are orthogonal to the level curves y=yx defined implicitly by fx,y=c, where c is a real constant.
Represent the level curve in the position-vector form R=xy(x) so a vector tangent to this curve is then R′x=1y′(x).
By implicitly differentiating fx,yx≡c to get fx+fy y′=0, the derivative y′x=−fx/fy is obtained.
Hence, the tangent vector is R′x=1−fx/fy, and ∇f·R′x=fxfy·1−fx/fy = fx−fx=0.
The gradient ∇f is therefore orthogonal to the level curve yx defined implicitly by fx,y=c.
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