I'm following the 18.02 course of MIT of multivariate calculus. Topic is directional derivatives.
Here is the question:
The function $T = x^2+2y^2+2z^2 $ gives the temperature at each point in space. At point $P=(1,1,1)$, in which direction should you go to get the most rapid decrease in $T$ What is the directional derivative in this direction?
Here is part of the answer:
We know that the fastest increase is in the direction of $\nabla T = \langle 2x,4y,4z \rangle $. At $P$, the fastest decrease is in the direction of $-\nabla T|_P= -\langle 1,2,2\rangle $. The unit vector in this direction is $\hat u = -\langle 1/3,2/3,2/3 \rangle$. The rate of change in this direction is $-|\nabla T|= -3$.
Here is what I don't understand:
It seems that there is a factor $1/2$ to obtain $-\nabla T|_P= -\langle1,2,2\rangle$ from $\nabla T = \langle 2x,4y,4z \rangle$. Where does it come from?