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.
<< Previous Example Section 4.5
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)