Show that the vector field F = y e^xyi+x e^xyj+0k is irrotational.

Question

Show that the vector field

[latex]\mathbf{F}=y\mathbf{e}^{xy}\mathbf{i}+x\mathbf{e}^{xy}\mathbf{j}+0\mathbf{k}[/latex]

is irrotational.

Accepted Answer

[latex]\nabla \times F = \begin{vmatrix} i & j & k \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ ye^{xy} & xe^{xy} & 0 \end{vmatrix}\ [/latex]

[latex]=\left({\frac{\partial}{\partial y}}0-{\frac{\partial}{\partial z}}x\,\mathrm{e}^{x y}\right)\mathrm{i}-\left({\frac{\partial}{\partial x}}0-{\frac{\partial}{\partial z}}y\,\mathrm{e}^{x y}\right)\mathrm{j}+\left({\frac{\partial}{\partial x}}x\,\mathrm{e}^{x y}-{\frac{\partial}{\partial y}}y\,\mathrm{e}^{x y}\right)\mathrm{k}\[/latex]

[latex]=0\mathbf{i}+0\mathbf{j}+((x y\mathbf{e}^{x y}+\mathbf{e}^{x y})-(y x\mathbf{e}^{x y}+\mathbf{e}^{x y}))\mathbf{k}[/latex]

=0 for all x, y and z

The field is therefore irrotational.

Question 26.8: Show that the vector field F = y e^xyi+x e^xyj+0k is irrotat......

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