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Determine whether or not $F$ is a conservative vector field. If it is, find a function $f$ such that F=\(\newlineabla f\). (If the vector field is not conservative, enter DNE. $\newline$ $\begin{cases} F(x,y)=e^{x}\sin(y)i+e^{x}\cos(y)j \end{cases}$ , $f(x,y)=$

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Find derivatives of sine and cosine functions

Full solution

Q. Determine whether or not

F

is a conservative vector field. If it is, find a function

f

such that F=\(\newlineabla f\). (If the vector field is not conservative, enter DNE.

\newline

\begin{cases} F(x,y)=e^{x}\sin(y)i+e^{x}\cos(y)j \end{cases}

,

f(x,y)=

Check Curl of F: Determine if F is conservative by checking if the curl of F is zero. $\newline$ Curl F = $(\frac{\partial}{\partial y}(e^x \sin(y)) - \frac{\partial}{\partial x}(e^x \cos(y)))k$ $\newline$ = $(e^x \cos(y) - e^x \cos(y))k$ $\newline$ = $0k$
Verify Conservativity: Since the curl of $F$ is zero, $F$ is a conservative vector field.
Find Potential Function: Find a potential function $f$ such that F = \(\newlineabla f\). We need $\frac{\partial f}{\partial x} = e^x \sin(y)$ and $\frac{\partial f}{\partial y} = e^x \cos(y)$ .
Integrate $\frac{\partial f}{\partial x}$ : Integrate $\frac{\partial f}{\partial x} = e^x \sin(y)$ with respect to $x$ .
$f(x, y) = \int e^x \sin(y) \, dx$
$\quad\quad= e^x \sin(y) + g(y)$ , where $g(y)$ is a function of $y$ only.
Differentiate to Find $g'(y)$ : Differentiate $f(x, y) = e^x \sin(y) + g(y)$ with respect to $y$ to find $g'(y)$ .
$\frac{\partial}{\partial y}(e^x \sin(y) + g(y)) = e^x \cos(y) + g'(y)$
Since $\frac{\partial f}{\partial y} = e^x \cos(y)$ , we have $g'(y) = 0$ .
Integrate $g'(y)$ : Integrate $g'(y) = 0$ to find $g(y)$ . $\newline$ $g(y) = C$ , where $C$ is a constant.
Substitute $g(y)$ : Substitute $g(y)$ back into $f(x, y)$ . $f(x, y) = e^x \sin(y) + C$

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