The mean value of the cube, Force Field Laplace equation

[Arman777]

Homework Statement

I have a value of $$ U=U_0+x (∂U/∂x)+y(∂U/∂y)+z (∂U/∂z)+1/2x^2(∂^2U/∂x^2)+1/2y^(2∂^2U/∂y^2)+...$$

We need to find the mean value of the U. So the answer is

$$\overline{\rm U}\approx U_0+a^2/24(∇^2U)$$

Homework Equations

$$\overline{\rm U}=1/a^3 \int \int\int Udxdydz$$

The Attempt at a Solution

The problem I get is that I have to proof that,

$$K=1/a^3 \int \int\int x (∂U/∂x)+y(∂U/∂y)+z (∂U/∂z)dxdydz=0$$ but

$$L=1/a^3 \int \int\int 1/2x^2(∂^2U/∂x^2)+1/2y^(2∂^2U/∂y^2)+1/2z^(2∂^2U/∂z^2)=a^2/24(∇^2U)$$ but

I couldn't proceed why these are true.

The integral limits are from $-a/2$ to $a/2$

 

[haruspex]

Homework Statement

I have a value of $$ U=U_0+x (∂U/∂x)+y(∂U/∂y)+z (∂U/∂z)+1/2x^2(∂^2U/∂x^2)+1/2y^(2∂^2U/∂y^2)+...$$

I think you have to be careful interpreting that. The derivatives should be evaluated at 0, so for the purposes of the integration they are constants.
I.e. $\frac{\partial U}{\partial x}|_{x=0}$ etc.
Also, you have dropped a power of 2 on the y in the last term above.