Temperature of the point $(x,y,z)$ in space is measured by the formula $T(x,y,z) = e^{-x^{2}-2y^{2}-3z^{2}}$ . At the present an object is located at the point (1,1,1) and needs to be brought to a cooler location. What is the direction of fastest cooling down? What is the rate of cooling in this direction? Which direction should the object move so that the rate of cooling is $\sqrt{14e^{-12}}$ ?
I first find out the gradient and its magnitude. They should be the rate and direction of fastest cooling down, right? However for the second part of the question, i try to find the point $(a,b,c)$ for that direction but i get an imaginary number which doesn't make sense. Can someone give me some ideas?
If you check the problem set online again, there was a correction. The object should move so that the rate of cooling should be $e^{-6}\sqrt(14)$.