\( =r^{m} \) where \( r^{2}=x^{2}+y^{2}+z^{2} \), show that \( \frac{\partial^{2} V}{\partial x^{2}}+\frac{\partial^{2} V}{\partial y^{2}}+\frac{\partial^{2} V}{\partial z^{2}}=m(m+1) r^{m-2} \)

To show that \[ \frac{\partial^{2} V}{\partial x^{2}} + \frac{\partial^{2} V}{\partial y^{2}} + \frac{\partial^{2} V}{\partial z^{2}} = m(m+1) r^{m-2} \] where \( V = r^m \) and \( r^2 = x^2 + y^2 + z^2 \), we will use the chain rule and the properties of partial derivatives. First, we express \( r \) in terms of \( x, y, z \): \[ r = \sqrt{x^2 + y^2 + z^2} \] Now, we can compute the first derivatives of \( V \): \[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x}(r^m) = m r^{m-1} \frac{\partial r}{\partial x} \] Next, we need to find \( \frac{\partial r}{\partial x} \): \[ \frac{\partial r}{\partial x} = \frac{x}{r} \] Thus, we have: \[ \frac{\partial V}{\partial x} = m r^{m-1} \cdot \frac{x}{r} = m r^{m-2} x \] Similarly, we can find the derivatives with respect to \( y \) and \( z \): \[ \frac{\partial V}{\partial y} = m r^{m-2} y \] \[ \frac{\partial V}{\partial z} = m r^{m-2} z \] Now, we compute the second derivatives. Starting with \( \frac{\partial^2 V}{\partial x^2} \): \[ \frac{\partial^2 V}{\partial x^2} = \frac{\partial}{\partial x} \left( m r^{m-2} x \right) \] Using the product rule: \[ \frac{\partial^2 V}{\partial x^2} = m \left( \frac{\partial}{\partial x}(r^{m-2}) x + r^{m-2} \frac{\partial x}{\partial x} \right) \] Now we need \( \frac{\partial r^{m-2}}{\partial x} \): \[ \frac{\partial r^{m-2}}{\partial x} = (m-2) r^{m-3} \frac{\partial r}{\partial x} = (m-2) r^{m-3} \cdot \frac{x}{r} = (m-2) r^{m-4} x \] Substituting this back into our expression for \( \frac{\partial^2 V}{\partial x^2} \): \[ \frac{\partial^2 V}{\partial x^2} = m \left( (m-2) r^{m-4} x^2 + r^{m-2} \right) \] Now, we can write: \[ \frac{\partial^2 V}{\partial x^2} = m (m-2) r^{m-4} x^2 + m r^{m-2} \] Next, we compute \( \frac{\partial^2 V}{\partial y^2} \) and \( \frac{\partial^2 V}{\partial z^2} \) in a similar manner: \[ \frac{\partial^2 V}{\partial y^2} = m (m-2) r^{m-4} y^2 + m r^{m-2} \] \[ \frac{\partial^2 V}{\partial z^2} = m (m-2) r^{m-4} z^2 + m r^{m-2} \] Now, we sum these second derivatives: \[ \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = m (m-2) r^{m-4} (x^2 + y^2 + z^2) + 3m r^{m-2} \] Since \( x^2 + y^2 + z^2 = r^2 \), we can substitute this into our equation: \[ = m (m-2) r^{m-4} r^2 + 3m r^{m-2} \] \[ = m (m-2) r^{m-2}