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Differential Operators#

A differential operator is an operator.

Gradient \(\grad f\) Rotation \(\rot \vec f\)
Divergenz \(\div \vec f\) Laplace \(\Delta\, f = \Sp \ma H_f(\vec x)\)


\(\mathrm{grad}\; f: \R \rightarrow \R^3\)

\[\grad f = \vec\nabla f = \vect{\frac{\partial f}{\partial x_1} \\ \vdots \\ \frac{\partial f}{\partial x_n} }\]


Der Gradient zeigt an, wie groß und in welcher Richtung die größte Steigung in einem Punkt ist.


\[\rot \vec{f} = \vec\nabla \times \vec f = \vect{\frac{\partial}{\partial x_1} \\ \dot : \\ \frac{\partial}{\partial x_n} } \times \vect{ f_1 \\ \dot : \\ f_n }\]


Die Rotation zeigt an, wie stark und um welche Achse sich ein Schaufelrad in einem Punkt drehen würde.


\[\div f = \vec\nabla \cdot \vec f = \vect{\frac{\partial}{\partial x_1} \\ \vdots \\ \frac{\partial}{\partial x_n} } \cdot \vect{ f_1 \\ \vdots \\ f_n}\]

Die Divergenz gibt an, wie stark (Strömungs-) Vektoren in einem Punkt auseinander gehen.

Example: Consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.


\[\underset{\nabla \cdot (\nabla f)}{\nabla^2} f = \vect{\frac{\partial}{\partial x_1} \\ \vdots \\ \frac{\partial}{\partial x_n} } \cdot \vect{\frac{\partial f}{\partial x_1} \\ \vdots \\ \frac{\partial f}{\partial x_n} }\]


Der Laplace-Operator zeigt an, wie stark die Krümmung des Skalarfeldes in einem Punkt ist.