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1. Sup (To, Yo 20

1. Sup (To, Yo 20

Advanced Engineering Mathematics
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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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.
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 Fx,y,z=k and x0,y0,z0∈S

Let Px0,y0,z0 be any point on the level surface Fx,y,z=k.

To show: gradient ∇Fx0,y0,z0 is perpendicular to the tangent plane to S at x0,y0,z0.

That is, ∇F 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 rt=xt,yt,zt be a curve on the level surface with rt0=x0,y0,z0.

Let gt=Fxt,yt,zt.

Since the curve is on the level surface,

gt=Fxt,yt,zt=c

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