Question

if grand,F=0,then F is called​

Answer 1

Answer:

The curl of a gradient is zero

 

Let f(x,y,z) be a scalar-valued function. Then its gradient

∇f(x,y,z)=(∂f∂x(x,y,z),∂f∂y(x,y,z),∂f∂z(x,y,z))

is a vector field, which we denote by F=∇f. We can easily calculate that the curl of F is zero.

We use the formula for curlF in terms of its components

curlF=(∂F3∂y−∂F2∂z,∂F1∂z−∂F3∂x,∂F2∂x−∂F1∂y).

Since each component of F is a derivative of f, we can rewrite the curl as

curl∇f=(∂2f∂y∂z−∂2f∂z∂y,∂2f∂z∂x−∂2f∂x∂z,∂2f∂x∂y−∂2f∂y∂x).

If f is twice continuously differentiable, then its second derivatives are independent of the order in which the derivatives are applied. All the terms cancel in the expression for curl∇f, and we conclude that curl∇f=0.