Calculate Line Integral for a Function on a Level Surface | Homework Equations

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Homework Help Overview

The problem involves calculating a line integral of the gradient of a function defined on a level surface, specifically where the function f(x,y,z) equals 5. Participants are exploring the implications of this setup on the integral and the properties of the gradient.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the relationship between the level surface and the values of the function at points p and q on the curve. Questions arise regarding the implications of the endpoints lying on the level surface and the behavior of the gradient.

Discussion Status

The discussion includes confirmations of reasoning regarding the values of f at points p and q, as well as clarifications about the nature of the gradient on a level surface. Some participants express confusion about the properties of the gradient in this context.

Contextual Notes

There is an ongoing exploration of the assumptions related to the function's behavior on the level surface and the implications for the gradient, with some participants questioning their understanding of these concepts.

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Homework Statement



Let f(x,y,z) be a function of three variables. Suppose that C is an oriented curve lying on the level surface f(x,y,z) = 5. Find the integral grad f dot dr.

Homework Equations





The Attempt at a Solution



integral grad f dot dr = integral f(q) - f(p)

not sure what to do
 
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If a point p is on the level surface f(x,y,z) = 5, what does that tell you about f(p)?
 
If the curve you are integrating over lies on the surface f(x,y,z)=5, then surely the endpoints p and q do as well right?...So f(p)=___? and f(q)=___?
 
f(P) - f(Q) = 5 - 5 = 0
 
is that correct
 
It's correct.
 
If f(x,y,z) = 5, how is grad f not equal to (0,0,0)? Jeez, today's just not my day.
 
f(x,y,z) = 5 on the level surface, not everywhere. If it was equal to 5 everywhere, grad f would be zero.
 
Oh yeah, duh. I think I need to drink more water or something.
 

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