Answer:
[tex]D_{\vec{v}}V(6,6,5)=48[/tex]
Step-by-step explanation:
You have the following potential function:
[tex]V(x,y,z)=5x^2-3xy+xyz[/tex] (1)
To find the rate of change of the potential at the point P(6,6,5) in the direction of v = i + j - k, you use the following formula:
[tex]D_{\vec{v}}V(x,y,z)=\bigtriangledown V(x,y,z)\cdot \vec{v}[/tex] (2)
First, you calculate the gradient of V:
[tex]\bigtriangledown V(x,y,z)=\frac{\partial}{\partial x}V(x,y,z)\hat{i}+\frac{\partial}{\partial y}V(x,y,z)\hat{i}+\frac{\partial}{\partial z}V(x,y,z)\hat{i}\\\\\bigtriangledown V(x,y,z)=(10x-3y+yz)\hat{i}+(-3x+xz)\hat{j}+(xy)\hat{k}\\\\\bigtriangledown V(6,6,5)=(10(6)-3(6)+(6)(5))\hat{i}+(-3(6)+(6)(5))\hat{j}+((6)(6))\hat{k}\\\\\bigtriangledown V(6,6,5)=72\hat{i}+12\hat{j}+36\hat{k}[/tex]
Next, you replace in the equation (2):
[tex]D_{\vec{v}}V(6,6,5)=(72\hat{i}+12\hat{j}+36\hat{k})\cdot(\hat{i}+\hat{j}-\hat{k})\\\\D_{\vec{v}}V(6,6,5)=48[/tex]
Then, the rate of change of the potential at the point P(6,6,5) in the direction of v, is 48.
