Question

Suppose that over a certain region of space the electrical potential V is given by the following equation. $V(x, y, z) = 3x^2 - 3xy + xyz$ (a) Find the rate of change of the potential at $P(5, 2, 5)$ in the direction of the vector $\mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k}$. (b) In which direction does $V$ change most rapidly at $P$? (c) What is the maximum rate of change at $P$?

Name: suppose that over a certain region of space the electrical potential v is given by the following equation vx y z 3x2 3xy xyz a find the rate of change of the potential at p5 2 5 in the direc 36843
Uploaded: 2022-08-13T00:21:48-08:00
Duration: 3 min 25 s
Channel: Antonio Lim
Description: suppose that over a certain region of space the electrical potential v is given by the following equation vx y z 3x2 3xy xyz a find the rate of change of the potential at p5 2 5 in the direc 36843

          Suppose that over a certain region of space the electrical potential V is given by the following equation.
$V(x, y, z) = 3x^2 - 3xy + xyz$
(a) Find the rate of change of the potential at $P(5, 2, 5)$ in the direction of the vector $\mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k}$.
(b) In which direction does $V$ change most rapidly at $P$?
(c) What is the maximum rate of change at $P$?

suppose that over a certain region of space the electrical potential v is given by the following equation vx y z 3x2 3xy xyz a find the rate of change of the potential at p5 2 5 in the direc 36843

Added by Vicente L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Antonio Lim

08/13/2022

Step 1

$\nabla V = \left(\frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z}\right)$ $\frac{\partial V}{\partial x} = 6x - 3y + yz$ $\frac{\partial V}{\partial y} = -3x + xz$ $\frac{\partial V}{\partial z} = xy$ So, $\nabla V = (6x Show more…

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Suppose that over a certain region of space the electrical potential V is given by the following equation. $V(x, y, z) = 3x^2 - 3xy + xyz$ (a) Find the rate of change of the potential at $P(5, 2, 5)$ in the direction of the vector $\mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k}$. (b) In which direction does $V$ change most rapidly at $P$? (c) What is the maximum rate of change at $P$?