Question

Determine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y, z) = 7y2z3 i + 14xyz3 j + 21xy2z2 k

Answer 1

Solution:

Given: F(x, y, z) = 7y²z³ i + 14xyz³ j + 21xy²z² k

F is a conservative field if curl F = 0, there exist a scalar potential function  such that F = $\triangledown f$.

curl f = $= \left | \begin{matrix} i & j & k\\  \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\  7y^{2}z^{3} & 14xyz^{3} & 21xy^{2}z^{2} \end{matrix} \right |$

$= i\left [ \frac{\partial }{\partial y}\left ( 21xy^{2}z^{2} \right ) - \frac{\partial }{\partial z}\left ( 14xyz^{3} \right ) \right ] - j \left [ \frac{\partial }{\partial x}\left ( 21xy^{2}z^{2} \right ) - \frac{\partial }{\partial z}\left ( 7y^{2}z^{3} \right ) \right ] + k\left [ \frac{\partial }{\partial x}\left ( 14xyz^{3} \right ) - \frac{\partial }{\partial y}\left ( 7y^{2}z^{3} \right ) \right ]$

$= i \left ( 0 \right ) - j\left ( 0 \right ) + k\left ( 0 \right )$

$= 0$

Thus,curl F is conservative.

To find f such that F = gradf.

$7y²z³ i + 14xyz³ j + 21xy²z² k = i\left ( \frac{\mathrm{d}f }{\mathrm{d} x} \right ) + j \left ( \frac{\mathrm{d}f }{\mathrm{d} y} \right ) + k \left ( \frac{\mathrm{d}f }{\mathrm{d} z} \right )$

$\frac{\mathrm{d}f }{\mathrm{d} x} = 7y²z³  $

$\frac{\mathrm{d}f }{\mathrm{d} x} = 14xyz³$

$\frac{\mathrm{d}f }{\mathrm{d} x} = 21xy²z²$

Integrate the above terms with respect to x.

$f = 7xy²z³ + c_{1}$

$f = 7xy²z³ + c_{2}$

$f = 7xy²z³ + c_{3}$

Hence, the scalar potential function is $f = 7xy²z³ + c$.