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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 $$(\sqrt{15},\sqrt{10},5),$$ 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. $$ v*\nabla T(\sqrt{15},\sqrt{10},5)$$ 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!

Qurultay
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Martin Pham
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  • What, exactly, is your question? – amd Apr 10 '19 at 22:18
  • If the fly's speed is 2, what rate of temperature change does it experience at t=0? – Martin Pham Apr 10 '19 at 23:09
  • 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? – amd Apr 10 '19 at 23:14
  • 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. – Martin Pham Apr 10 '19 at 23:16
  • OK. Why do you think your answer is incorrect? – amd Apr 10 '19 at 23:24
  • I'm getting my problems through an online platform, where I submit my answers digitally. (Just the final answer, no calculation). I'm able to "verify" my answers that I submit, and the platform will tell me if my answer is correct or not, but will not show the correct answer. – Martin Pham Apr 10 '19 at 23:28
  • Perhaps it wants the answer in a different form. Have you tried eliminating the radical from the denominator? – amd Apr 10 '19 at 23:35
  • I did. From previous problems; form does not really matter, as long as the values are the same. Thanks for the suggestion though. – Martin Pham Apr 10 '19 at 23:39
  • FWIW, your answer matches mine. The only other thing I can suggest at this point is to double-check that you’ve copied the problem itself correctly. – amd Apr 10 '19 at 23:39

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