Question

Determine the direction of maximum increase of the scalar field f(x,y,z)=xe^y+z^2 at the point O( 1,ln2,3).

Answer 1

Answer

Gradient of g at P₀

$\nabla g|_{P_0}=\begin{pmatrix} e^y\\ xe^y\\ 2z \end{pmatrix}_{(1, ln2, 3) }$

$=\begin{pmatrix} 2\\ 2\\ 2/3 \end{pmatrix}$

The direction in which g increase most rapidly

$u=\frac{3}{\sqrt{19}}\begin{pmatrix} 1 \\ 1\\ 1/3 \end{pmatrix}$