Answer to Question #214798 in Vector Calculus for jannathul prudous

Question

prove : div cur l f =0

Answer 1

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 $\vec F$

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