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Solenoidal and irrotational vector field
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[QUOTE="Dominika, post: 5466694, member: 593312"] [h2]Homework Statement [/h2] I am to prove (using the equations for gradient, divergence and curl in spherical polar coordinates) that vector field $$\mathbf{w}=w_{\psi}(r,\theta)\hat e_{\psi}$$ is solenoidal, find $$w_{\psi}(r,\theta)$$ when it's irrotational and find a potential in this case. [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] For vector field to be solenoidal, divergence should be zero, so I get the equation: $$\nabla\cdot\mathbf{w}=\frac{1}{r\sin\theta}\frac{\partial w_{\psi}(r,\theta)}{\partial \psi}=0$$ For a vector field to be irrotational, the curl has to be zero. After substituting values into equation, I get: $$\cos\theta\cdot w_{\psi}+\frac{\partial w_{\psi}}{\partial \theta}\cdot \sin\theta=0$$ and $$w_{\psi}+\frac{\partial w_{\psi}}{\partial r}\cdot r=0$$. Is it right? How to proceed? [/QUOTE]
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Solenoidal and irrotational vector field
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