Gradients of harmonic functions

[musicmar]

Homework Statement

a. show that delφ=div(gradφ) for any function φ.
b. show that φ is harmonic if and only if div(gradφ)=o.
c. Show that if F is the gradient of a harmonic function, then curl(F)=0 and div(F)=0.
d. Show F=<xz,-yz,1/2(x²-y²)> is the gradient of a harmonic function. What is the flux of F through a closed surface?

The Attempt at a Solution

I did parts (a) and (b), but am now stuck on (c) (and thus (d)). Can someone explain to me what the gradient of a harmonic function is and how you find one? Thanks.

 

[ystael]

"$$\mathbf{F}$$ is the gradient of a harmonic function" just means that there exists a harmonic function $$\varphi$$ ($$\nabla^2\varphi = 0$$) such that $$\mathbf{F} = \nabla \varphi$$. For part (c), you just need to compute $$\nabla \times \mathbf{F}$$ and $$\nabla \cdot \mathbf{F}$$ based on this assumption. For part (d), you need to find a suitable harmonic function $$\varphi$$.

 

[musicmar]

Why do I need to compute ∇xF and ∇·F? I'm sorry; I'm still confused about what this will tell me and how this relates to φ.

 

[ystael]

Part (c) asks you: if $$\mathbf{F}$$ is the gradient of a harmonic function, what are $$\nabla\times\mathbf{F}$$ and $$\nabla\cdot\mathbf{F}$$? This means your hypothesis is that there exists some harmonic function $$\varphi$$ such that $$\mathbf{F} = \nabla\varphi$$. You don't know, or need to know, anything about $$\varphi$$ other than those two facts: that it is harmonic, and that its gradient is $$\mathbf{F}$$. From these facts you can draw the conclusions you need, by rewriting $$\nabla\times\mathbf{F}$$ and $$\nabla\cdot\mathbf{F}$$ in terms of $$\varphi$$.