Given a surface $z=f(x,y)$ we need to find the change in temperature $t(x,y,z)$ in the direction of $(a,b)$ at point $(x_0,y_0)$.
My current way of thinking is finding the tangent plane of $f(x,y)$ at $(x_0,y_0)$ using $h = f(x_0, y_0) + f_x(x_0, y_0) * (x-x_0) + f_y(x_0, y_0) * (y-y_0)$
I then proceed to find the directional vector by doing: $v = (x_0+a, y_0+b,h(x_0+a, y_0+b)) - (x_0, y_0, f(x_0, y_0))$
And normalizing it:
$v = v/||v||$
I then dot the gradient of t with v.
My question is how correct is this and if not where have I gone wrong ?
EDIT:: Fixed a mistake