Question 27.8: The vector field v is derivable from the potential φ = 2xy +......

Engineering Mathematics A Foundation for Electronic, Electrical, Communications and Systems Engineers [1040294]

The vector field $\mathbf{v}$ is derivable from the potential $\phi = 2xy + zx$ . Find $\mathbf{v}$

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If $\mathbf{v}$ is derivable from the potential $\phi$ , then $\mathbf{v}=\nabla \phi$ and so

\mathbf{v}=\nabla \phi=(2 y+z) \mathbf{i}+2 x \mathbf{j}+x \mathbf{k}

This vector field is conservative as is easily verified by finding curl $\mathbf{v}$ . In fact,

\begin{aligned} \nabla \times \mathbf{v} & =\left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 2 y+z & 2 x & x \end{array}\right| \\ & =0 \mathbf{i}-(1-1) \mathbf{j}+(2-2) \mathbf{k} \\ & =\mathbf{0} \end{aligned}

Indeed, recall from Example 26.12 that curl (grad φ) is identically zero for any $\phi$ .

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