Question #214798

prove : div cur l f =0


1
Expert's answer
2021-07-09T06:40:02-0400

We introduce the vector differential operator \nabla (“del”) as


=ix+jy+kz\nabla=\vec i \dfrac{\partial}{\partial x}+\vec j \dfrac{\partial}{\partial y}+\vec k \dfrac{\partial}{\partial z}

Then for the vector field F\vec F


curlF=×Fcurl\vec F=\nabla \times\vec F

divF=Fdiv\vec F=\nabla \cdot\vec F

The cross product ×F\nabla \times\vec F  is perpendicular to both \nabla and F.\vec F.

Hence for any vector field F\vec F

(×F)=0\nabla\cdot(\nabla\times\vec F)=0

Therefore for any vector field F\vec F

div(curlF)=(×F)=0div(curl \vec F)=\nabla\cdot(\nabla\times\vec F)=0


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