Suppose that the temperature at a point $(x,y,z)$ in space is given by $T(x,y,z)=\frac{80}{1+x^2+2y^2+3z^2}$ where $T$ is measured in degrees celsius and $x$,$y$ and $z$ in meters.
In which direction does the temperature increase fastest at the point $(1,1,-2)$? What is the maximum rate of increase?
I first had to find the vector field which is $\operatorname{grad}T=-5i/8-5j/4+15k/4$ and assume it be the direction and then determine the magnitude as the rate. I need a quick critic on this and please help to make a critical decision on how make corrections. Thank you in advance
Best Answer
You compute the partial derivatives at the point you are given. You get
$$(\partial_{x}T,\partial_{y}T,\partial_{z}T)=-160\left(\frac{x}{(1+x^{2}+2y^{2}+3z^{2})^{2}},\frac{2y}{(1+x^{2}+2y^{2}+3z^{2})^{2}},\frac{3z}{(1+x^{2}+2y^{2}+3z^{2})^{2}}\right)$$
that in $(1,1,-2)$ is equal to
$$-\frac{5}{8}(1,2,-6)=\left(-\frac{5}{8},-\frac{5}{4},\frac{15}{4}\right)$$
Then the direction is that given by the above vector. After normalization it is approximately $\left(-\frac{5}{32},-\frac{5}{16},\frac{15}{16}\right)$. The modulus of this vector is $1.00098$, so the approximation is pretty precise. Your reasoning is totally correct.