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.
<< Previous Example Section 4.5
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. 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)