Question

Obtain the directional derivative for a scalar field phi( X,Y,Z)= 3x^2y-y^3z^2 at the points (1,-2,-1) in the direction i+j+k

Answer 1

directional derivative is

$$\vec{A}=\nabla\phi=\frac{∂ \phi}{∂x}i+\frac{∂ \phi}{∂y}j+\frac{∂ \phi}{∂z}k$$

then

$\frac{∂ \phi}{∂x}=6xy=-12$

$\frac{∂ \phi}{∂y}=3x^2-3y^2z^2=3-12=-9$

$\frac{∂ \phi}{∂z}=2y^3z=16$

Answer

$$\vec{A}=12i-9j+16k$$