Evaluate total derivative of $f(x, y, z ) = x^2 + y^2 -z^2$ at $a=(1,-2,1)$ what is it's value in the direction of $h = (-1,1,1)$.
I'm not sure, I completely understand how to do this. Is it just:
$Df(a)(h) =\frac{\partial f}{\partial x}(a_1)\cdot h_1 + \frac{\partial f}{\partial y}(a_2)\cdot h_2 + \frac{\partial f}{\partial z}(a_3)\cdot h_3 = -1(2) + 1(-4) + 1(-2) = -8$
$\frac{\partial f}{\partial x} = 2x, \frac{\partial f}{\partial y} =2y,\frac{\partial f}{\partial z} = -2z$
Almost. Say we had to look at it in the direction of $h=(-10,10,10)$. This is the same direction as before -- the answer should be the same. But, if you carry through your method, you'd get an answer tenfold of what it should be. The problem is that you have to normalize $h$ first, by dividing by $||h||$, to get an answer that is $\sqrt{3}$ times smaller than your one.