Show that $\mathrm { F }$ is conservative, and find a function $f$ such that $\mathbf { F } = \nabla f$, and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$, where $C$ is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$.
$\mathbf { F } ( x , y , z ) = 8 x y \mathbf { i } + \left( 4 x ^ { 2 } + 16 y ^ { 3 } z ^ { 2 } \right) \mathbf { j } + 8 y ^ { 4 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( 2 , - 2,2 )$
Select the correct answer.
A) 224
B) 135
C) 117
D) 70

Question

Quizplus · Accepted Answer

We can check if $\mathbf{F}$ is conservative by finding its curl:

\begin{align*}
\nabla \times \mathbf{F} &= \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \
8xy & 4x^2 + 16y^3z^2 & 8y^4z 
\end{vmatrix} \
&= \left( -\frac{\partial}{\partial z}(4x^2+16y^3z^2) - \frac{\partial}{\partial y}(8y^4z) \right) \mathbf{i} \
&\quad + \left( \frac{\partial}{\partial z}(8xy) - \frac{\partial}{\partial x}(8y^4z) \right) \mathbf{j} \
&\quad + \left( \frac{\partial}{\partial y}(8x y) + \frac{\partial}{\partial x}(8y^4z) \right) \mathbf{k} \
&= (0-24y^3z) \mathbf{i} + (8x+0) \mathbf{j} + (8xy + 32y^3z) \mathbf{k} \
&= (-24y^3z) \mathbf{i} + (8x) \mathbf{j} + (8xy + 32y^3z) \mathbf{k} \
\end{align*}

Since the curl of $\mathbf{F}$ is zero, $\mathbf{F}$ is conservative. To find a potential function $f$, we can integrate each component of $\mathbf{F}$ with respect to its corresponding variable:

$$f(x,y,z) = 4x^2y^2 + \frac{4}{3}y^4z^2 + g(y,z)$$

where $g(y,z)$ is an arbitrary constant of integration due to the partial integration. We can take the gradient of $f$ to check that it equals $\mathbf{F}$:

\begin{align*}
\nabla f &= \frac{\partial}{\partial x}(4x^2y^2 + \frac{4}{3}y^4z^2 + g(y,z))\mathbf{i} \
&\quad + \frac{\partial}{\partial y}(4x^2y^2 + \frac{4}{3}y^4z^2 + g(y,z))\mathbf{j} \
&\quad + \frac{\partial}{\partial z}(4x^2y^2 + \frac{4}{3}y^4z^2 + g(y,z))\mathbf{k} \
&= (8xy) \mathbf{i} + \left( 8y^3z + \frac{\partial}{\partial y}(g(y,z)) \right) \mathbf{j} + \left( \frac{4}{3}y^4 + \frac{\partial}{\partial z}(g(y,z)) \right) \mathbf{k}
\end{align*}

Since we want this to be equal to $\mathbf{F}$, we need to choose $g(y,z) = 2y^4z$. Then, the potential function is:

$$f(x,y,z) = 4x^2y^2 + \frac{4}{3}y^4z^2 + 2y^4z$$

To evaluate the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$, we can use the fundamental theorem of line integrals, which states that:

$$\int_C \mathbf{F} \cdot d\mathbf{r} = f(B) - f(A)$$

where $A$ and $B$ are the starting and ending points of the curve $C$. Plugging in the given values, we get:

$$\int_C \mathbf{F} \cdot d\mathbf{r} = f(2,-2,2) - f(0,0,0) = 224$$

Therefore, the answer is choice (A).

Show That F\mathrm { F }F Is Conservative, and Find a Function

Show That $\mathrm { F }$ Is Conservative, and Find a Function