Calculus: I can't understand why curl of gradient of a scalar is zero

[chingcx]

(Sorry, the title should read "...why curl of gradient of a scalar "function" is zero)

Of course I know how to compute curl, graident, divergence. Algebrically I know curl of gradient of a scalar function is zero.

But I want to know the reason behind this...and also the reason why gradient of divergence of a vector function is always zero.

This really makes me feeling bad for a long time. Thanks in advance.

 

[Defennder]

The gradient of a scalar function would always give a conservative vector field. Now think carefully about what curl is. If you've done an E&M course with vector calculus, think back to the time when the textbook (or your course notes) derived $$\nabla \times \mathbf{H} = \mathbf{J}$$ using Ampere's circuital law. What is the closed path integral of a conservative field?

I'm wondering about your second question too...

 

[atyy]

But I want to know the reason behind this...and also the reason why gradient of divergence of a vector function is always zero.

Is this true? In Gauss's law, the divergence of the electric field is to equal an arbitrary charge density. I would be surprised if the gradient of an arbitrary charge density is zero.

 

[Ben Niehoff]

$$\nabla(\nabla \cdot \vec F) = 0$$

is certainly false. I think you mean

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

which is true.

 

[tiny-tim]

(Sorry, the title should read "...why curl of gradient of a scalar "function" is zero)

Hi chingcx!

(curl grad f)_x = ∂/∂y(∂f/∂z) - ∂/∂z(∂A/∂y) = 0

Similarly, div curl A = 0

...and also the reason why gradient of divergence of a vector function is always zero.

But (grad div A)_x = ∂/∂x(∂A_x/∂x) + ∂/∂x(∂A_y/∂y) + ∂/∂x(∂A_x/∂z) ≠ 0

 

[Defennder]

Is this true? In Gauss's law, the divergence of the electric field is to equal an arbitrary charge density. I would be surprised if the gradient of an arbitrary charge density is zero.

Hey you're right. Wow I can't believe I didn't even bother thinking about whether it might be correct, as opposed to why it might be correct.

 

[chingcx]

$$\nabla(\nabla \cdot \vec F) = 0$$

is certainly false. I think you mean

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

which is true.

ya, sorry, I mean divergence of curl of vector function is always zero. Why is that true?

 

[NoMoreExams]

Does this help you http://en.wikipedia.org/wiki/Solenoidal_vector_field