Question

Determine the direction of maximum increase of the scalar field

f(x,y,z) = xe^y + z²

at the point O(1,ln2,3)

Answer 1

The direction of maximum increase of scalar field is defined by it's gradient :

$\vec\nabla f=\begin{pmatrix} \frac{\partial}{\partial x}f \\ \frac{\partial}{\partial y}f \\ \frac{\partial}{\partial z}f \end{pmatrix}$

Let's calculate it :

$\vec\nabla f=\begin{pmatrix} e^y \\ xe^y \\ 2z \end{pmatrix}$

Therefore the gradient at the point O is :

$\vec\nabla f(O) = \begin{pmatrix} 2 \\ 2 \\ 6 \end{pmatrix}$

This vector defines the diretion of the fastest growth of f at the point O.