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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.) $\newline$ \(\mathbf{F}(x,y)=\langle y e^{x}+\sin(y), e^{x}+x \cos(y) \rangle

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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

\(\mathbf{F}(x,y)=\langle y e^{x}+\sin(y), e^{x}+x \cos(y) \rangle

Identify components of vector field: Identify the components of the vector field $F(x, y)$ . $F(x, y) = (y e^x + \sin(y))\mathbf{i} + (e^x + x \cos(y))\mathbf{j}$ .
Check for conservativity: Check if $F$ is conservative by comparing the partial derivatives of the components. $\newline$ For $F$ to be conservative, $\frac{\partial P}{\partial y}$ must equal $\frac{\partial Q}{\partial x}$ , where $P = y e^x + \sin(y)$ and $Q = e^x + x \cos(y)$ .
Calculate $\frac{\partial P}{\partial y}$ : Calculate $\frac{\partial P}{\partial y}$ . $\frac{\partial P}{\partial y} = \frac{\partial}{\partial y} (y e^x + \sin(y)) = e^x + \cos(y)$ .
Calculate $\frac{\partial Q}{\partial x}$ : Calculate $\frac{\partial Q}{\partial x}$ . $\newline$ $\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x} (e^x + x \cos(y)) = e^x - x \sin(y)$ .

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