Question: Let \(R\) be a region in \(\mathbb{R}^{3}\) with smooth boundary and \(f,g\) be smooth real-valued functions on \(R\). Suppose that \(\Delta f=0\) inside \(R\) and \(f=g\) on \(\partial R\).

1. Prove that
   \[\int_{R}\|\nabla f\|^{2} d V=\int_{R} \nabla f \cdot \nabla g d V\]
2. The following 'Cauchy inequality' is true for any vector functions \(\mathrm{F}\) and \(\mathrm{G}\).
   \[\left(\int_{R} \mathbf{F} \cdot \mathbf{G} d V\right)^{2} \leq\left(\int_{R}\|\mathbf{F}\|^{2} d V\right)\left(\int_{R}\|\mathbf{G}\|^{2} d V\right)\]
   Assuming this, prove that
   \[\int_{R}\|\nabla f\|^{2} d V \leq \int_{R}\|\nabla g\|^{2} d V\]
3. Explain how this inequality might relate to Dirichlet's principle: Among all the functions with the same values on \(\partial R\), the harmonic function has the least energy.
4. Let \(B\) be the unit ball in \(\mathbb{R}^{3}\) centered at the origin. Compute

Justify your answer using the previous parts. Hint: the solution should be a line or two at most.

Price: $2.99

Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document