1. Sup (To, Yo 20
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question

Transcribed Image Text:1. Suppose S is a level surface with equation F(x,y, z) =k and
(To, Yo 20) E S.
VFTo Yor Zo) is perpendicular to the tangent plane to S at (ro, Yo zo)
Suppose VF(xo, Yo, Zo) = 0. Show that the gradient
02
0.
Expert Solution

Step 1
Given S is a level surface with equation and
Let be any point on the level surface .
To show: gradient is perpendicular to the tangent plane to S at .
That is, at P is perpendicular to the surface.
This would imply that it is perpendicular to the tangent to any curve that lies on the surface and passes through P.
Using the chain rule,
Let be a curve on the level surface with .
Let
Since the curve is on the level surface,
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