Determine whether the vector field [latex]\mathrm{F}(x, y)=-y e^{-x y} \mathrm{i}-x e^{-x y} \mathrm{j}[/latex] is conservative.

With [latex]P=-y e^{-x y}[/latex] and [latex]Q=-x e^{-x y}[/latex], we find [latex]\frac{\partial P}{\partial y}=x y e^{-x y}-e^{-x y}=\frac{\partial Q}{\partial x} .[/latex] The components of [latex]\mathrm{F}[/latex] are continuous and have continuous partial derivatives. Thus (6) [latex]\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}[/latex] (6) holds throughout the [latex]x y[/latex]-plane, which is a simply connected region. From the converse in Theorem 9.9.4 we can conclude that [latex]\mathrm{F}[/latex] is conservative.

Question 9.9.6: Determine whether the vector field F(x,y) = -ye^-xyi - xe^-x......

Advanced Engineering Mathematics [1367744]

Determine whether the vector field $\mathrm{F}(x, y)=-y e^{-x y} \mathrm{i}-x e^{-x y} \mathrm{j}$ is conservative.

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With $P=-y e^{-x y}$ and $Q=-x e^{-x y}$ , we find

$\frac{\partial P}{\partial y}=x y e^{-x y}-e^{-x y}=\frac{\partial Q}{\partial x} .$

The components of $\mathrm{F}$ are continuous and have continuous partial derivatives. Thus (6)

$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$ (6)

holds throughout the $x y$ -plane, which is a simply connected region. From the converse in Theorem 9.9.4 we can conclude that $\mathrm{F}$ is conservative.