Suppose that over a certain region of space the electrical potential $V$ is given by $V ( x , y , z ) = 8 x ^ { 2 } - 7 x y + 7 x y z$.
Find the rate of change of the potential at $( - 1,1 , - 1 )$ in the direction of the vector $\mathbf { v } = 7 \mathbf { i } + 10 \mathbf { j } - 8 \mathbf { k }$.
Select the correct answer.
A) $- 15.099$
B) $14.569856$
C) 44
D) 20
E) $- 0.96$

Question

Quizplus · Accepted Answer

The rate of change of potential in the direction of vector $\mathbf{v}$ is given by the dot product of the gradient of $V$ at the point of interest and the unit vector in the direction of $\mathbf{v}$:
$$
abla V \cdot \frac{\mathbf{v}}{\left| \mathbf{v} ight|}$$
where
$$
abla V = \left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} ight)$$
is the gradient of $V$.

To find $
abla V$, we take partial derivatives of $V$ with respect to $x$, $y$, and $z$:
$$\frac{\partial V}{\partial x} = 16x - 7y + 7yz,$$
$$\frac{\partial V}{\partial y} = -7x + 7xz,$$
$$\frac{\partial V}{\partial z} = 7xy.$$
Thus,
$$
abla V = \left( 16x - 7y + 7yz, -7x + 7xz, 7xy ight).$$

At the point $( - 1,1 , - 1 )$, we have
$$
abla V = \left( -23, -14, -7 ight).$$

The unit vector in the direction of $\mathbf{v}$ is
$$\frac{\mathbf{v}}{\left| \mathbf{v} ight|} = \frac{1}{15} (7, 10, -8).$$

Thus, the rate of change of $V$ at $( - 1,1 , - 1 )$ in the direction of $\mathbf{v}$ is
$$
abla V \cdot \frac{\mathbf{v}}{\left| \mathbf{v} ight|} = (-23) \cdot \frac{7}{15} + (-14) \cdot \frac{10}{15} + (-7) \cdot \frac{-8}{15} = \boxed{-0.96}.$$

Suppose That Over a Certain Region of Space the Electrical VVV

Suppose That Over a Certain Region of Space the Electrical $V$