Determine whether or not F is a conservative vector field. If it is, find a function f such that F=grad f. (If the vector field is not conservative, enter DNE. {:[F(x","y)=e^(x)sin(y)i+e^(x)cos(y)j^(˙)],[f(x","y)=]:}

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Determine whether or not  F is a conservative vector field. If it is, find a function  f such that  F=grad f. (If the vector field is not conservative, enter DNE.  {:[F(x","y)=e^(x)sin(y)i+e^(x)cos(y)j^(˙)],[f(x","y)=]:}

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Check Curl of F:  Determine if F is conservative by checking if the curl of F is zero. Curl F = (∂/∂y(e^x sin(y)) - ∂/∂x(e^x cos(y)))k        = (e^x cos(y) - e^x cos(y))k        = 0kVerify 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 = grad f. We need ∂f/∂x = e^x sin(y) and ∂f/∂y = e^x cos(y).Integrate ∂f/∂x:  Integrate ∂f/∂x = e^x sin(y) with respect to x. f(x, y) = ∫e^x sin(y) dx         = 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). ∂/∂y(e^x sin(y) + g(y)) = e^x cos(y) + g'(y) Since ∂f/∂y = e^x cos(y), we have g'(y) = 0.Integrate g'(y):  Integrate g'(y) = 0 to find g(y). 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

Determine whether or not $F$ is a conservative vector field. If it is, find a function $f$ such that $F=f$ . (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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