To prove that [tex]\( V_{xx} + V_{yy} + V_{zz} = f''(r) + \frac{2}{r} f'(r) \)[/tex] given [tex]\( V = f(r) \)[/tex] and [tex]\( r^2 = x^2 + y^2 + z^2 \)[/tex], we follow these steps:
1. Define [tex]\( r \)[/tex] in terms of [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[
r^2 = x^2 + y^2 + z^2 \implies r = \sqrt{x^2 + y^2 + z^2}
\][/tex]
2. Express [tex]\( V \)[/tex] as a function of [tex]\( r \)[/tex]:
[tex]\[
V = f(r)
\][/tex]
3. Find the first and second derivatives of [tex]\( r \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\frac{\partial r}{\partial x} = \frac{x}{r}
\][/tex]
[tex]\[
\frac{\partial^2 r}{\partial x^2} = \frac{r - x \frac{\partial r}{\partial x}}{r^2} = \frac{r - \frac{x^2}{r}}{r^2} = \frac{r^2 - x^2}{r^3} = \frac{y^2 + z^2}{r^3}
\][/tex]
4. Use the chain rule to find the first and second derivatives of [tex]\( V \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
V_x = \frac{\partial V}{\partial x} = \frac{d f}{d r} \cdot \frac{\partial r}{\partial x} = f'(r) \cdot \frac{x}{r}
\][/tex]
[tex]\[
V_{xx} = \frac{\partial}{\partial x} \left( f'(r) \cdot \frac{x}{r} \right) = f''(r) \cdot \left( \frac{x}{r} \right)^2 + f'(r) \cdot \frac{r^2 - x^2}{r^3} = f''(r) \frac{x^2}{r^2} + f'(r) \cdot \frac{y^2 + z^2}{r^3}
\][/tex]
5. Find similar expressions for the second derivatives with respect to [tex]\( y \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[
V_{yy} = f''(r) \frac{y^2}{r^2} + f'(r) \cdot \frac{x^2 + z^2}{r^3}
\][/tex]
[tex]\[
V_{zz} = f''(r) \frac{z^2}{r^2} + f'(r) \cdot \frac{x^2 + y^2}{r^3}
\][/tex]
6. Sum the second partial derivatives to compute the Laplacian [tex]\( V_{xx} + V_{yy} + V_{zz} \)[/tex]:
[tex]\[
V_{xx} + V_{yy} + V_{zz} = f''(r) \left( \frac{x^2 + y^2 + z^2}{r^2} \right) + f'(r) \cdot \left( \frac{y^2 + z^2 + x^2 + z^2 + x^2 + y^2}{r^3} \right)
\][/tex]
Simplify the terms:
[tex]\[
\frac{x^2 + y^2 + z^2}{r^2} = 1
\][/tex]
[tex]\[
\frac{2(x^2 + y^2 + z^2)}{r^3} = \frac{2r^2}{r^3} = \frac{2}{r}
\][/tex]
7. Combine these results to get the final expression:
[tex]\[
V_{xx} + V_{yy} + V_{zz} = f''(r) \cdot 1 + f'(r) \cdot \frac{2}{r} = f''(r) + \frac{2}{r} f'(r)
\][/tex]
Hence, we have proven that:
[tex]\[
V_{xx} + V_{yy} + V_{zz} = f''(r) + \frac{2}{r} f'(r)
\][/tex]
