Question

a) Determine the direction of maximum increase of the scalar field
2 f (x,y,z) xe z y   at the point O1,ln2,3.

Answer 1

Answer:

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}∇f=⎝⎛∂x∂f∂y∂f∂z∂f⎠⎞

Let's calculate it :

\vec\nabla f=\begin{pmatrix} e^y \\ xe^y \\ 2z \end{pmatrix}∇f=⎝⎛eyxey2z⎠⎞

Therefore the gradient at the point O is :

\vec\nabla f(O) = \begin{pmatrix} 2 \\ 2 \\ 6 \end{pmatrix}∇f(O)=⎝⎛226⎠⎞

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

Explanation:

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Answer 2

Answer:

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