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gradient rotation divergence laplace

Definition

A differential operator is an operator.

Gradient gradf\text{grad}f Rotation rot๐Ÿ\text{rot}\boldsymbol f
Divergenz div๐Ÿ\text{div}\boldsymbol f Laplace ฮ”f=Sp๐‡f(๐ฑ)\Delta\, f = \,\text{Sp}\boldsymbol{H}_f(\boldsymbol x)

Gradient

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

gradf=โˆ‡f=(โˆ‚fโˆ‚x1โ‹ฎโˆ‚fโˆ‚xn)\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 groรŸ und in welcher Richtung die grรถรŸte Steigung in einem Punkt ist.

Rotation

rot๐Ÿ=โˆ‡ร—๐Ÿ=(โˆ‚โˆ‚x1:ฬ‡โˆ‚โˆ‚xn)ร—(f1:ฬ‡fn)\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 wรผrde.

Divergenz

divf=โˆ‡โ‹…๐Ÿ=(โˆ‚โˆ‚x1โ‹ฎโˆ‚โˆ‚xn)โ‹…(f1โ‹ฎfn)\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 (Strรถmungs-) 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 Krรผmmung des Skalarfeldes in einem Punkt ist.

References