Also ich habe folgendes gerechnet;
Gradient \(\nabla f\):
Der Gradient für \( f(x) \) ist def. als:
\(\nabla f(x) = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n} \right)\)
Für \( f(x) = |x| \):
\(\frac{\partial f}{\partial x_i} = \frac{\partial |x|}{\partial x_i}\)
\( |x| = \sqrt{x_1^2 + x_2^2 + \ldots + x_n^2} \), ist \( \frac{\partial |x|}{\partial x_i} = \frac{x_i}{|x|} \).
=> \(\nabla f(x) = \left( \frac{x_1}{|x|}, \frac{x_2}{|x|}, \ldots, \frac{x_n}{|x|} \right)\)
Laplace-Operator \(\Delta f\);
\(\Delta f(x) = \sum_{i=1}^{n} \frac{\partial^2 f}{\partial x_i^2}\)
Für \( f(x) = |x| \):
\(\frac{\partial^2 f}{\partial x_i^2} = \frac{\partial}{\partial x_i} \left( \frac{x_i}{|x|} \right)\)
zweite Ableitung:
\(\frac{\partial}{\partial x_i} \left( \frac{x_i}{|x|} \right) = \frac{\partial}{\partial x_i} \left( x_i (x_1^2 + x_2^2 + \ldots + x_n^2)^{-\frac{1}{2}} \right)\)
Produktregel:
\(\frac{\partial}{\partial x_i} \left( x_i (x_1^2 + x_2^2 + \ldots + x_n^2)^{-\frac{1}{2}} \right) = (x_1^2 + x_2^2 + \ldots + x_n^2)^{-\frac{1}{2}} - x_i^2 (x_1^2 + x_2^2 + \ldots + x_n^2)^{-\frac{3}{2}}\)
Also...
\(\frac{\partial^2 f}{\partial x_i^2} = \left( \frac{x_1^2 + x_2^2 + \ldots + x_n^2 - 2x_i^2}{|x|^3} \right)\)
Laplace-Operator:
\(\Delta f(x) = \sum_{i=1}^{n} \frac{\partial^2 f}{\partial x_i^2} = \sum_{i=1}^{n} \left( \frac{x_1^2 + x_2^2 + \ldots + x_n^2 - 2x_i^2}{|x|^3} \right)\)
