(Solved): For Laplace's equation the following Theorems are valid, prove them: Theorem 1: If \( \phi_{1}, \cd ...

For Laplace's equation the following Theorems are valid, prove them: Theorem 1: If \( \phi_{1}, \cdots, \phi_{n} \) are solutions of \( \nabla^{2} \phi=0 \) \[ \Rightarrow \phi=c_{1} \phi_{1}+\cdots+c_{n} \phi_{n} \] is also a solution of the same equation with \( c_{i} \) arbitrary constants. Theorem 2 (uniqueness): Two solutions of the Laplace's equation satisfying the same boundary conditions differ by at most one additive constant.