Answer to Question #214798 in Vector Calculus for jannathul prudous

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


"\\nabla=\\vec i \\dfrac{\\partial}{\\partial x}+\\vec j \\dfrac{\\partial}{\\partial y}+\\vec k \\dfrac{\\partial}{\\partial z}"

Then for the vector field "\\vec F"


"curl\\vec F=\\nabla \\times\\vec F"

"div\\vec F=\\nabla \\cdot\\vec F"

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

Hence for any vector field "\\vec F"

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

Therefore for any vector field "\\vec F"

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


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