Answer to Question #275799 in Calculus for Hafsa Tahsin

Question #275799

[SADT3] For scalar functions u and v, show that



B=( nabla u)*( nabla v)



is solenoidal and that



A= 1 2 (u nabla v-v nabla u)



is a vector potential for B, i.e. B= nabla* A

1
Expert's answer
2021-12-06T13:28:41-0500

B=(u)×(v)B=( \nabla u)\times( \nabla v)

A=12(uvvu)A= \frac{1}{2} (u \nabla v-v \nabla u)

B=×AB= \nabla\times A


for solenoidal vector field:

B=0\nabla \cdot B=0


u=uxi+uyj+uzk\nabla u=\frac{\partial u}{\partial x}i+\frac{\partial u}{\partial y}j+\frac{\partial u}{\partial z}k


v=vxi+vyj+vzk\nabla v=\frac{\partial v}{\partial x}i+\frac{\partial v}{\partial y}j+\frac{\partial v}{\partial z}k


B=(u)×(v)=ijkuxuyuzvxvyvz=B=( \nabla u)\times( \nabla v)=\begin{vmatrix} i & j&k \\ \frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}&\frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}&\frac{\partial v}{\partial z}\\ \end{vmatrix}=


=(uyvzvyuz)i(uxvzuzvx)j+(uxvyuyvx)k=0=(\frac{\partial u}{\partial y}\frac{\partial v}{\partial z}-\frac{\partial v}{\partial y}\frac{\partial u}{\partial z})i-(\frac{\partial u}{\partial x}\frac{\partial v}{\partial z}-\frac{\partial u}{\partial z}\frac{\partial v}{\partial x})j+(\frac{\partial u}{\partial x}\frac{\partial v}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x})k=0


so, B=0\nabla \cdot B=0



A=12(uvvu)=12(uvxi+uvyj+uvzkvuxivuyjvuzk)A= \frac{1}{2} (u \nabla v-v \nabla u)=\frac{1}{2} (u\frac{\partial v}{\partial x}i+u\frac{\partial v}{\partial y}j+u\frac{\partial v}{\partial z}k-v\frac{\partial u}{\partial x}i-v\frac{\partial u}{\partial y}j-v\frac{\partial u}{\partial z}k)


×A=[y(uvzvuz)z(uvyvuy)]i+[z(uvzvuz)x(uvzvuz)]j+\nabla\times A=[\frac{\partial}{\partial y}(u\frac{\partial v}{\partial z}-v\frac{\partial u}{\partial z})-\frac{\partial}{\partial z}(u\frac{\partial v}{\partial y}-v\frac{\partial u}{\partial y})]i+[\frac{\partial}{\partial z}(u\frac{\partial v}{\partial z}-v\frac{\partial u}{\partial z})-\frac{\partial}{\partial x}(u\frac{\partial v}{\partial z}-v\frac{\partial u}{\partial z})]j+


+[x(uvyvuy)y(uvxvux)]k=B+[\frac{\partial}{\partial x}(u\frac{\partial v}{\partial y}-v\frac{\partial u}{\partial y})-\frac{\partial}{\partial y}(u\frac{\partial v}{\partial x}-v\frac{\partial u}{\partial x})]k=B


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