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The temperature in 3-space is given by:

$$ T(x,y,z)= \frac12(2x^2+5y^2+4z^2) $$

At time $$t = 0,$$ a fly passes through the point $$(\sqrt15,\sqrt10,5),$$ flying along the curve of intersection of the surfaces

$$ z = x^2-y^2 $$ and $$ z^2 = x^2+y^2 $$

If the fly's speed is 2, what rate of temperature change does it experience at t=0?

So far, I've tried:

  1. Finding the normals for the surfaces
  2. Finding a tangent vector to the intersection in the point given
  3. Finding the fly's velocity vector, v
  4. $$ v*\nabla T(\sqrt15,\sqrt10,5)$$ Picture of my quick calculations (excuse the handwriting)

Thank you for any help.

Edit: To clarify: The work I've done gave me an incorrect answer; so I'm looking for any input on what I've done wrong and/or what the correct answer to my problem would be. Cheers!

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  • $\begingroup$ What, exactly, is your question? $\endgroup$ – amd Apr 10 at 22:18
  • $\begingroup$ If the fly's speed is 2, what rate of temperature change does it experience at t=0? $\endgroup$ – Martin Pham Apr 10 at 23:09
  • $\begingroup$ No, that’s the statement of the problem that you’re been given to solve. You’ve shown all of your work for it. Are you asking for someone to verify your solution? $\endgroup$ – amd Apr 10 at 23:14
  • $\begingroup$ I should've clarified, but basically, yeah. I have no idea if the work I've done is correct, just wanted to show that I've actually tried solving the problem myself (and gotten the wrong answer). So looking for someone to verify if I'm on the right track and/or help me with the initial problem. $\endgroup$ – Martin Pham Apr 10 at 23:16
  • $\begingroup$ OK. Why do you think your answer is incorrect? $\endgroup$ – amd Apr 10 at 23:24

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