Exercise 44

Suppose that the temperature in a region of space is described by $$\begin{gathered} T(x,y,z) = 100e^{-x^2-y^2-z^2} \end{gathered}$$ and that you are standing at the point $(1,2,-1)$.

1. Find the instantaneous rate of change of the temperature in the direction of $\vv=\langle 0, 1, 2\rangle$ at the point $(1,2,-1)$. Remember that you should first find a unit vector in the direction of $\vv$.

2. In what direction from the point $(1,2,-1)$ would you move to cause the temperature to decrease as quickly as possible?

3. How fast does the temperature decrease in this direction?

4. Find a direction in which the temperature does not change at $(1,2,-1)$.