- #1
- 2,168
- 193
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##
Last edited: