1. **Problem statement:** Given $V = r^m$ where $r^2 = x^2 + y^2 + z^2$, prove that $$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = m(m+1)r^{m-2}.$$ 2. **Recall:** $r = \sqrt{x^2 + y^2 + z^2}$, so $V = r^m = (x^2 + y^2 + z^2)^{m/2}$. 3. **First derivatives:** Using the chain rule, $$\frac{\partial V}{\partial x} = m r^{m-1} \frac{\partial r}{\partial x} = m r^{m-1} \frac{x}{r} = m x r^{m-2}.$$ Similarly, $$\frac{\partial V}{\partial y} = m y r^{m-2}, \quad \frac{\partial V}{\partial z} = m z r^{m-2}.$$ 4. **Second derivatives:** Differentiate again, $$\frac{\partial^2 V}{\partial x^2} = \frac{\partial}{\partial x} (m x r^{m-2}) = m \left( r^{m-2} + x \frac{\partial}{\partial x} r^{m-2} \right).$$ 5. Compute $\frac{\partial}{\partial x} r^{m-2}$: $$\frac{\partial}{\partial x} r^{m-2} = (m-2) r^{m-3} \frac{\partial r}{\partial x} = (m-2) r^{m-3} \frac{x}{r} = (m-2) x r^{m-4}.$$ 6. Substitute back: $$\frac{\partial^2 V}{\partial x^2} = m \left( r^{m-2} + x (m-2) x r^{m-4} \right) = m \left( r^{m-2} + (m-2) x^2 r^{m-4} \right).$$ 7. Similarly for $y$ and $z$: $$\frac{\partial^2 V}{\partial y^2} = m \left( r^{m-2} + (m-2) y^2 r^{m-4} \right),$$ $$\frac{\partial^2 V}{\partial z^2} = m \left( r^{m-2} + (m-2) z^2 r^{m-4} \right).$$ 8. **Sum all second derivatives:** $$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = m \left( 3 r^{m-2} + (m-2) r^{m-4} (x^2 + y^2 + z^2) \right).$$ 9. Since $x^2 + y^2 + z^2 = r^2$, substitute: $$= m \left( 3 r^{m-2} + (m-2) r^{m-4} r^2 \right) = m \left( 3 r^{m-2} + (m-2) r^{m-2} \right) = m (3 + m - 2) r^{m-2} = m (m+1) r^{m-2}.$$ **Final answer:** $$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = m(m+1) r^{m-2}.$$