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gradient rotation divergence laplace

Definition

A differential operator is an operator.

Gradient ${\text{grad}}f$ Rotation ${\text{rot}}{\boldsymbol f}$
Divergenz ${\text{div}}{\boldsymbol f}$ Laplace $\Delta\, f = {\,\text{Sp}}{\boldsymbol{H}}_f({\boldsymbol x})$

Gradient

$\mathrm{grad}\; f: {\mathop{\mathbb R}}\rightarrow {\mathop{\mathbb R}}^3$

$${\text{grad}}f = {\boldsymbol \nabla} f = {\begin{pmatrix} \frac{\partial f}{\partial x_1} \\ \vdots \\ \frac{\partial f}{\partial x_n} \end{pmatrix}}$$

Explanation

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

Rotation

$${\text{rot}}{\boldsymbol f} = {\boldsymbol \nabla} \times {\boldsymbol f} = {\begin{pmatrix} \frac{\partial}{\partial x_1} \\ \dot : \\ \frac{\partial}{\partial x_n} \end{pmatrix}} \times {\begin{pmatrix} f_1 \\ \dot : \\ f_n \end{pmatrix}}$$

Explanation

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

Divergenz

$${\text{div}}f = {\boldsymbol \nabla} \cdot {\boldsymbol f} = {\begin{pmatrix} \frac{\partial}{\partial x_1} \\ \vdots \\ \frac{\partial}{\partial x_n} \end{pmatrix}} \cdot {\begin{pmatrix} f_1 \\ \vdots \\ f_n \end{pmatrix}}$$

Explanation

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.

Laplace

$${\tiny\begin{matrix}\\ \normalsize \nabla^2 \\[1em] \scriptsize \nabla \cdot (\nabla f) \end{matrix}} f = {\begin{pmatrix} \frac{\partial}{\partial x_1} \\ \vdots \\ \frac{\partial}{\partial x_n} \end{pmatrix}} \cdot {\begin{pmatrix} \frac{\partial f}{\partial x_1} \\ \vdots \\ \frac{\partial f}{\partial x_n} \end{pmatrix}}$$

Explanation

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

References

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