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:
- Finding the normals for the surfaces
- Finding a tangent vector to the intersection in the point given
- Finding the fly's velocity vector, v
- $$ v*\nabla T(\sqrt15,\sqrt10,5)$$ 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!