Formelsammlung Tensoranalysis

g r a d (f) = f_{, i} {\hat{e}}_{i}

g r a d (\vec{f}) = {\vec{f}}_{, i} \otimes {\hat{e}}_{i} = {\hat{e}}_{i} \otimes g r a d (f_{i}) = f_{i, j} {\hat{e}}_{i} \otimes {\hat{e}}_{j}

g r a d (f) = f_{, ρ} {\hat{e}}_{ρ} + \frac{f_{, φ}}{ρ} {\hat{e}}_{φ} + f_{, z} {\hat{e}}_{z}

\begin{matrix} g r a d (\vec{f}) = & {\hat{e}}_{ρ} \otimes g r a d (f_{ρ}) + {\hat{e}}_{φ} \otimes g r a d (f_{φ}) + {\hat{e}}_{z} \otimes g r a d (f_{z}) \\ + \frac{1}{ρ} (f_{ρ} {\hat{e}}_{φ} - f_{φ} {\hat{e}}_{ρ}) \otimes {\hat{e}}_{φ} \end{matrix}

#Krummlinige Koordinaten:

g r a d (f) = f_{, r} {\hat{e}}_{r} + \frac{f_{, ϑ}}{r} {\hat{e}}_{ϑ} + \frac{f_{, φ}}{r \sin (ϑ)} {\hat{e}}_{φ}

\begin{matrix} g r a d (\vec{f}) = & {\hat{e}}_{r} \otimes g r a d (f_{r}) + {\hat{e}}_{ϑ} \otimes g r a d (f_{ϑ}) + {\hat{e}}_{φ} \otimes g r a d (f_{φ}) \\ + \frac{f_{r}}{r} (𝟏 - {\hat{e}}_{r} \otimes {\hat{e}}_{r}) - {\hat{e}}_{r} \otimes \frac{f_{ϑ} {\hat{e}}_{ϑ} + f_{φ} {\hat{e}}_{φ}}{r} + \frac{f_{ϑ} {\hat{e}}_{φ} - f_{φ} {\hat{e}}_{ϑ}}{r \tan (ϑ)} \otimes {\hat{e}}_{φ} \end{matrix}

Christoffelsymbole: $Γ_{i j}^{k} = {\vec{g}}_{i, j} \cdot {\vec{g}}^{k}$

Vektorfelder:

g r a d ({\vec{g}}_{i}) = Γ_{i j}^{k} {\vec{g}}_{k} \otimes {\vec{g}}^{j}

g r a d ({\vec{g}}^{k}) = - Γ_{i j}^{k} {\vec{g}}^{i} \otimes {\vec{g}}^{j}

g r a d (f^{i} {\vec{g}}_{i}) = {f^{i} |}_{j} {\vec{g}}_{i} \otimes {\vec{g}}^{j}

g r a d (f_{i} {\vec{g}}^{i}) = {f_{i} |}_{j} {\vec{g}}^{i} \otimes {\vec{g}}^{j}

Mit den kovarianten Ableitungen

{f^{i} |}_{j} = f_{, j}^{i} + Γ_{k j}^{i} f^{k}

{f_{i} |}_{j} = f_{i, j} - Γ_{i j}^{k} f_{k}

Tensorfelder:

g r a d (𝐓) = 𝐓_{, k} \otimes {\vec{g}}^{k}

Für ein Tensorfeld zweiter Stufe:

\begin{matrix} g r a d (T^{i j} {\vec{g}}_{i} \otimes {\vec{g}}_{j}) = & {T_{i j} |}_{k} {\vec{g}}^{i} \otimes {\vec{g}}^{j} \otimes {\vec{g}}^{k}, {T_{i j} |}_{k} & = T_{i j, k} - Γ_{i k}^{l} T_{l j} - Γ_{j k}^{l} T_{i l} \\ g r a d (T^{i j} {\vec{g}}_{i} \otimes {\vec{g}}_{j}) = & {T^{i j} |}_{k} {\vec{g}}_{i} \otimes {\vec{g}}_{j} \otimes {\vec{g}}^{k}, {T^{i j} |}_{k} & = T_{, k}^{i j} + Γ_{l k}^{i} T^{l j} + Γ_{l k}^{j} T^{i l} \\ g r a d (T_{i}^{. j} {\vec{g}}^{i} \otimes {\vec{g}}_{j}) = & {T_{i}^{. j} |}_{k} {\vec{g}}^{i} \otimes {\vec{g}}_{j} \otimes {\vec{g}}^{k}, {T_{i}^{. j} |}_{k} & = T_{i, k}^{. j} - Γ_{i k}^{l} T_{l}^{. j} + Γ_{l k}^{j} T_{i}^{. l} \\ g r a d (T_{. j}^{i}) {\vec{g}}_{i} \otimes {\vec{g}}^{j} = & {T_{. j}^{i} |}_{k} {\vec{g}}_{i} \otimes {\vec{g}}^{j} \otimes {\vec{g}}^{k}, {T_{. j}^{i} |}_{k} & = T_{. j, k}^{i} + Γ_{l k}^{i} T_{. j}^{l} - Γ_{j k}^{l} T_{. l}^{i} \end{matrix}

Produktregel für Gradienten

\begin{matrix} g r a d (f g) & = & (f_{, i} g + f g_{, i}) {\hat{e}}_{i} & = & g r a d (f) g + f g r a d (g) \\ g r a d (f \vec{g}) & = & (f_{, i} \vec{g} + f {\vec{g}}_{, i}) \otimes {\hat{e}}_{i} & = & \vec{g} \otimes g r a d (f) + f g r a d (\vec{g}) \\ g r a d (\vec{f} \cdot \vec{g}) & = & ({\vec{f}}_{, i} \cdot \vec{g} + \vec{f} \cdot {\vec{g}}_{, i}) {\hat{e}}_{i} & = & \vec{g} \cdot g r a d (\vec{f}) + \vec{f} \cdot g r a d (\vec{g}) \\ = & g r a d (\vec{f})^{⊤} \cdot \vec{g} + g r a d (\vec{g})^{⊤} \cdot \vec{f} \\ g r a d (\vec{f} \times \vec{g}) & = & ({\vec{f}}_{, i} \times \vec{g} + \vec{f} \times {\vec{g}}_{, i}) \otimes {\hat{e}}_{i} & = & \vec{f} \times g r a d (\vec{g}) - \vec{g} \times g r a d (\vec{f}) \end{matrix}

In drei Dimensionen ist speziell^[2]

g r a d (\vec{f} \cdot \vec{g}) = g r a d (\vec{f}) \cdot \vec{g} + g r a d (\vec{g}) \cdot \vec{f} + \vec{f} \times r o t (\vec{g}) + \vec{g} \times r o t (\vec{f})

Beliebige Basis:

g r a d (f_{i} {\vec{b}}_{i}) = {\vec{b}}_{i} \otimes g r a d (f_{i}) + f_{i} g r a d ({\vec{b}}_{i})

Divergenz

Definition der Divergenz/Allgemeines

Vektorfeld $\vec{f}$ :

d i v (\vec{f}) = \nabla \cdot \vec{f} = S p (g r a d (\vec{f}))

d i v (\vec{x}) = S p (g r a d (\vec{x})) = S p (𝟏) = 3

Klassische Definition für ein Tensorfeld T:^[1]

d i v (𝐓) \cdot \vec{c} = d i v (𝐓^{⊤} \cdot \vec{c}) \forall \vec{c} \in 𝕍

→

d i v (𝐓) = \nabla \cdot (𝐓^{⊤})

Koordinatenfreie Darstellung:

Volumen $v$ mit
Oberfläche $a$ mit äußerem vektoriellem Oberflächenelement $d \vec{a}$

d i v (\vec{f}) = lim_{v \to 0} (\frac{1}{v} \int_{a} \vec{f} \cdot d \vec{a})

Zusammenhang mit den anderen Differentialoperatoren:

\begin{matrix} d i v (\vec{f}) & = & \nabla \cdot \vec{f} & = & S p (g r a d (\vec{f})) \\ d i v (f 𝟏) & = & \nabla \cdot (f 𝟏) & = & g r a d (f) \end{matrix}

Nützliche Formeln für Divergenzen

\begin{matrix} d i v (f (r) \vec{x}) = & 3 f (r) + r \frac{d f (r)}{d r} \end{matrix}

mit $r = | \vec{x} |$ ^[2.1]

Divergenz in verschiedenen Koordinatensystemen

d i v (\vec{f}) = {\vec{f}}_{, i} \cdot {\hat{e}}_{i} = f_{i, i}

d i v (𝐓) = 𝐓_{, i} \cdot {\hat{e}}_{i} = T_{i j, j} {\hat{e}}_{i}

\nabla \cdot 𝐓 = {\hat{e}}_{i} \cdot 𝐓_{, i} = T_{i j, i} {\hat{e}}_{j} = T_{j i, j} {\hat{e}}_{i}

d i v (\vec{f}) = \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ f_{ρ}) + \frac{1}{ρ} f_{φ, φ} + f_{z, z}

\begin{matrix} d i v (𝐓) = & (T_{ρ ρ, ρ} + \frac{1}{ρ} (T_{ρ φ, φ} + T_{ρ ρ} - T_{φ φ}) + T_{ρ z, z}) {\hat{e}}_{ρ} \\ + (T_{φ ρ, ρ} + \frac{1}{ρ} (T_{φ φ, φ} + T_{ρ φ} + T_{φ ρ}) + T_{φ z, z}) {\hat{e}}_{φ} \\ + (T_{z ρ, ρ} + \frac{1}{ρ} (T_{z φ, φ} + T_{z ρ}) + T_{z z, z}) {\hat{e}}_{z} \end{matrix}

$\nabla \cdot 𝐓 = d i v (𝐓^{⊤})$ ergibt sich hieraus durch Vertauschen von T_ab durch T_ba.

\begin{matrix} d i v (\vec{f}) = & f_{r, r} + \frac{2 f_{r} + f_{ϑ, ϑ}}{r} + \frac{f_{ϑ} \cos (ϑ) + f_{φ, φ}}{r \sin (ϑ)} \\ d i v (𝐓) = & (T_{r r, r} + \frac{2 T_{r r} - T_{ϑ ϑ} - T_{φ φ} + T_{r ϑ, ϑ}}{r} + \frac{T_{r φ, φ} + T_{r ϑ} \cos (ϑ)}{r \sin (ϑ)}) {\hat{e}}_{r} \\ (T_{ϑ r, r} + \frac{2 T_{ϑ r} + T_{r ϑ} + T_{ϑ ϑ, ϑ}}{r} + \frac{(T_{ϑ ϑ} - T_{φ φ}) \cos (ϑ) + T_{ϑ φ, φ}}{r \sin (ϑ)}) {\hat{e}}_{ϑ} \\ (T_{φ r, r} + \frac{2 T_{φ r} + T_{r φ} + T_{φ ϑ, ϑ}}{r} + \frac{(T_{ϑ φ} + T_{φ ϑ}) \cos (ϑ) + T_{φ φ, φ}}{r \sin (ϑ)}) {\hat{e}}_{φ} \end{matrix}

$\nabla \cdot 𝐓 = d i v (𝐓^{⊤})$ ergibt sich hieraus durch Vertauschen von T_ab durch T_ba.

Produktregel für Divergenzen

d i v (f \vec{g}) = \nabla \cdot (f \vec{g}) = (f_{, i} \vec{g} + f {\vec{g}}_{, i}) \cdot {\hat{e}}_{i} = g r a d (f) \cdot \vec{g} + f d i v (\vec{g})

d i v (\vec{f} \times \vec{g}) = \nabla \cdot (\vec{f} \times \vec{g}) = ({\vec{f}}_{, i} \times \vec{g} + \vec{f} \times {\vec{g}}_{, i}) \cdot {\hat{e}}_{i} = \vec{g} \cdot r o t (\vec{f}) - \vec{f} \cdot r o t (\vec{g})

\begin{matrix} d i v (\vec{f} \otimes \vec{g}) = & ({\vec{f}}_{, i} \otimes \vec{g} + \vec{f} \otimes {\vec{g}}_{, i}) \cdot {\hat{e}}_{i} & = & g r a d (\vec{f}) \cdot \vec{g} + d i v (\vec{g}) \vec{f} \\ d i v (f 𝐓) = & (f_{, i} 𝐓 + f 𝐓_{, i}) \cdot {\hat{e}}_{i} & = & 𝐓 \cdot g r a d (f) + f d i v (𝐓) \\ d i v (𝐓 \cdot \vec{f}) = & (𝐓_{, i} \cdot \vec{f} + 𝐓 \cdot {\vec{f}}_{, i}) \cdot {\hat{e}}_{i} & = & d i v (𝐓^{⊤}) \cdot \vec{f} + 𝐓^{⊤} : g r a d (\vec{f}) \\ d i v (\vec{f} \times 𝐓) = & ({\vec{f}}_{, i} \times 𝐓 + \vec{f} \times 𝐓_{, i}) \cdot {\hat{e}}_{i} & = & \vec{i} (g r a d (\vec{f}) \cdot 𝐓^{⊤}) + \vec{f} \times d i v (𝐓) \end{matrix}

\begin{matrix} \nabla \cdot (\vec{f} \otimes \vec{g}) = & {\hat{e}}_{i} \cdot ({\vec{f}}_{, i} \otimes \vec{g} + \vec{f} \otimes {\vec{g}}_{, i}) & = & (\nabla \cdot \vec{f}) \vec{g} + (\nabla \otimes \vec{g})^{⊤} \cdot \vec{f} \\ \nabla \cdot (f 𝐓) = & {\hat{e}}_{i} \cdot (f_{, i} 𝐓 + f 𝐓_{, i}) & = & (\nabla f) \cdot 𝐓 + f \nabla \cdot 𝐓 \\ \nabla \cdot (𝐓 \cdot \vec{f}) = & {\hat{e}}_{i} \cdot (𝐓_{, i} \cdot \vec{f} + 𝐓 \cdot {\vec{f}}_{, i}) & = & (\nabla \cdot 𝐓) \cdot \vec{f} + 𝐓 : (\nabla \otimes \vec{f}) \\ \nabla \cdot (𝐓 \times \vec{f}) = & {\hat{e}}_{i} \cdot (𝐓_{, i} \times \vec{f} + 𝐓 \times {\vec{f}}_{, i}) & = & (\nabla \cdot 𝐓) \times \vec{f} - \vec{i} ((\nabla \otimes \vec{f})^{⊤} \cdot 𝐓) \end{matrix}

Beliebige Basis:

d i v (f_{i} {\vec{b}}_{i}) = \nabla \cdot (f_{i} {\vec{b}}_{i}) = g r a d (f_{i}) \cdot {\vec{b}}_{i} + f_{i} d i v ({\vec{b}}_{i})

d i v (T^{i j} {\vec{b}}_{i} \otimes {\vec{b}}_{j}) = (g r a d (T^{i j}) \cdot {\vec{b}}_{j}) {\vec{b}}_{i} + T^{i j} (g r a d ({\vec{b}}_{i}) \cdot {\vec{b}}_{j} + d i v ({\vec{b}}_{j}) {\vec{b}}_{i})

\nabla \cdot (T^{i j} {\vec{b}}_{i} \otimes {\vec{b}}_{j}) = ((\nabla T^{i j}) \cdot {\vec{b}}_{i}) {\vec{b}}_{j} + T^{i j} ((\nabla \cdot {\vec{b}}_{i}) {\vec{b}}_{j} + (\nabla {\vec{b}}_{j}) \cdot {\vec{b}}_{i})

Produkt mit Konstanten:

d i v (f 𝐂) = 𝐂 \cdot g r a d (f) \to d i v (f 𝟏) = g r a d (f)

\nabla \cdot (f 𝐂) = g r a d (f) \cdot 𝐂 \to \nabla \cdot (f 𝟏) = \nabla f

\begin{matrix} d i v (𝐂 \cdot \vec{f}) = 𝐂^{⊤} : g r a d (\vec{f}) \to d i v (\vec{f}) = & d i v (𝟏 \cdot \vec{f}) = 𝟏 : g r a d (\vec{f}) \\ = & S p (g r a d (\vec{f})) \end{matrix}

Rotation

Definition der Rotation/Allgemeines

Vektorfeld $\vec{f}$ :

r o t (\vec{f}) = \nabla \times \vec{f}

Klassische Definition für ein Tensorfeld T:^[1]

r o t (𝐓) \cdot \vec{c} = r o t (𝐓^{⊤} \cdot \vec{c}) \forall \vec{c} \in 𝕍

→

r o t (𝐓) = \nabla \times (𝐓^{⊤})

Allgemeine Identitäten:

{𝐓 = 𝐓}^{⊤} \to S p (r o t (𝐓)) = S p (\nabla \times 𝐓) = 0

r o t (\vec{x}) = r o t (f (| \vec{x} |) \vec{x}) = \vec{0}

Integrabilitätsbedingung^[3]: Jedes divergenzfreie Vektorfeld ist die Rotation eines Vektorfeldes:

d i v (\vec{f}) = 0 \to \exists \vec{g} : \vec{f} = r o t (\vec{g})

.

Siehe auch #Satz über rotationsfreie Felder.

Koordinatenfreie Darstellung:

Volumen $v$ mit
Oberfläche $a$ mit äußerem vektoriellem Oberflächenelement $d \vec{a}$

r o t (\vec{f}) = - lim_{v \to 0} (\frac{1}{v} \int_{a} \vec{f} \times d \vec{a})

Zusammenhang mit den anderen Differentialoperatoren:

\begin{matrix} r o t (f \vec{c}) & = g r a d (f) \times \vec{c} \\ r o t (\vec{f}) & = - \vec{i} (g r a d (\vec{f})) = \vec{i} (\nabla \otimes \vec{f}) = \nabla \times \vec{f} \end{matrix}

Rotation in verschiedenen Koordinatensystemen

\begin{matrix} r o t (\vec{f}) = & {\hat{e}}_{i} \times {\vec{f}}_{, i} = f_{j, i} {\hat{e}}_{i} \times {\hat{e}}_{j} = ϵ_{i j k} f_{j, i} {\hat{e}}_{k} \\ = & (f_{3, 2} - f_{2, 3}) {\hat{e}}_{1} + (f_{1, 3} - f_{3, 1}) {\hat{e}}_{2} + (f_{2, 1} - f_{1, 2}) {\hat{e}}_{3} \end{matrix}

r o t (𝐓) = {\hat{e}}_{i} \times 𝐓_{, i}^{⊤} = {\hat{e}}_{i} \times T_{l j, i} {\hat{e}}_{j} \otimes {\hat{e}}_{l} = ϵ_{i j k} T_{l j, i} {\hat{e}}_{k} \otimes {\hat{e}}_{l}

r o t (\vec{f}) = \frac{f_{z, φ} - ρ f_{φ, z}}{ρ} {\hat{e}}_{ρ} + (f_{ρ, z} - f_{z, ρ}) {\hat{e}}_{φ} + \frac{f_{φ} + ρ f_{φ, ρ} - f_{ρ, φ}}{ρ} {\hat{e}}_{z}

r o t (𝐓) = {\hat{e}}_{ρ} \times (𝐓_{, ρ}^{⊤}) + \frac{1}{ρ} {\hat{e}}_{φ} \times (𝐓_{, φ}^{⊤}) + {\hat{e}}_{z} \times (𝐓_{, z}^{⊤})

\nabla \times 𝐓 = {\hat{e}}_{ρ} \times 𝐓_{, ρ} + \frac{1}{ρ} {\hat{e}}_{φ} \times 𝐓_{, φ} + {\hat{e}}_{z} \times 𝐓_{, z}

\begin{matrix} r o t (\vec{f}) = & \frac{f_{φ, ϑ} \sin (ϑ) + f_{φ} \cos (ϑ) - f_{ϑ, φ}}{r \sin (ϑ)} {\hat{e}}_{r} + (\frac{f_{r, φ}}{r \sin (ϑ)} - \frac{f_{φ} + r f_{φ, r}}{r}) {\hat{e}}_{ϑ} \\ + \frac{f_{ϑ} + r f_{ϑ, r} - f_{r, ϑ}}{r} {\hat{e}}_{φ} \end{matrix}

r o t (𝐓) = {\hat{e}}_{r} \times (𝐓_{, r}^{⊤}) + \frac{1}{r} {\hat{e}}_{ϑ} \times (𝐓_{, ϑ}^{⊤}) + \frac{1}{r \sin (ϑ)} {\hat{e}}_{φ} \times (𝐓_{, φ}^{⊤})

\nabla \times 𝐓 = {\hat{e}}_{r} \times 𝐓_{, r} + \frac{1}{r} {\hat{e}}_{ϑ} \times 𝐓_{, ϑ} + \frac{1}{r \sin (ϑ)} {\hat{e}}_{φ} \times 𝐓_{, φ}

Produktregel für Rotationen

\begin{matrix} r o t (f \vec{g}) & = {\hat{e}}_{i} \times (f_{, i} \vec{g} + f {\vec{g}}_{, i}) = g r a d (f) \times \vec{g} + f r o t (\vec{g}) \\ r o t (\vec{f} \times \vec{g}) & = {\hat{e}}_{i} \times ({\vec{f}}_{, i} \times \vec{g} + \vec{f} \times {\vec{g}}_{, i}) \\ = ({\hat{e}}_{i} \cdot \vec{g}) {\vec{f}}_{, i} - ({\hat{e}}_{i} \cdot {\vec{f}}_{, i}) \vec{g} + ({\hat{e}}_{i} \cdot {\vec{g}}_{, i}) \vec{f} - ({\hat{e}}_{i} \cdot \vec{f}) {\vec{g}}_{, i} \\ = g r a d (\vec{f}) \cdot \vec{g} - d i v (\vec{f}) \vec{g} + d i v (\vec{g}) \vec{f} - g r a d (\vec{g}) \cdot \vec{f} \\ = d i v (\vec{f} \otimes \vec{g}) - d i v (\vec{g} \otimes \vec{f}) = \nabla \cdot (\vec{g} \otimes \vec{f}) - \nabla \cdot (\vec{f} \otimes \vec{g}) \end{matrix}

\begin{matrix} r o t (\vec{f} \otimes \vec{g}) & = {\hat{e}}_{i} \times ({\vec{g}}_{, i} \otimes \vec{f} + \vec{g} \otimes {\vec{f}}_{, i}) & = & r o t (\vec{g}) \otimes \vec{f} - \vec{g} \times g r a d (\vec{f})^{⊤} \\ r o t (f 𝐓) & = {\hat{e}}_{k} \times (f_{, k} 𝐓^{⊤} + f 𝐓_{, k}^{⊤}) & = & g r a d (f) \times (𝐓^{⊤}) + f r o t (𝐓) \end{matrix}

$\begin{matrix} r o t (𝐓 \cdot \vec{f}) & = {\hat{e}}_{k} \times (𝐓_{, k} \cdot \vec{f} + 𝐓 \cdot {\vec{f}}_{, k}) \\ = r o t (𝐓^{⊤}) \cdot \vec{f} + \vec{i} ({\hat{e}}_{k} \otimes 𝐓 \cdot {\vec{f}}_{, k}) \\ = r o t (𝐓^{⊤}) \cdot \vec{f} - \vec{i} (𝐓 \cdot g r a d (\vec{f})) \\ r o t (\vec{f} \times 𝐓) & = - r o t ((𝐓^{⊤} \times \vec{f})^{⊤}) \\ = - \nabla \times (𝐓^{⊤} \times \vec{f}) \\ = - (\nabla \times 𝐓^{⊤}) \times \vec{f} + 𝐓^{⊤} # (\nabla \otimes \vec{f}) \\ = - r o t (𝐓) \times \vec{f} + {(𝐓 # g r a d (\vec{f}))}^{⊤} \end{matrix}$

\begin{matrix} \nabla \times (\vec{f} \otimes \vec{g}) & = {\hat{e}}_{i} \times ({\vec{f}}_{, i} \otimes \vec{g} + \vec{f} \otimes {\vec{g}}_{, i}) & = & (\nabla \times \vec{f}) \otimes \vec{g} - \vec{f} \times (\nabla \otimes (\vec{g}) \\ \nabla \times (f 𝐓) & = {\hat{e}}_{k} \times (f_{, k} 𝐓 + f 𝐓_{, k}) & = & (\nabla f) \times 𝐓 + f \nabla \times 𝐓 \end{matrix}

$\begin{matrix} \nabla \times (𝐓 \cdot \vec{f}) & = {\hat{e}}_{k} \times (𝐓_{, k} \cdot \vec{f} + 𝐓 \cdot {\vec{f}}_{, k}) \\ = (\nabla \times 𝐓) \cdot \vec{f} + \vec{i} ({\hat{e}}_{k} \otimes 𝐓 \cdot {\vec{f}}_{, k}) \\ = (\nabla \times 𝐓) \cdot \vec{f} - \vec{i} (𝐓 \cdot (\nabla \otimes \vec{f})^{⊤}) \\ \nabla \times (𝐓 \times \vec{f}) & = {\hat{e}}_{k} \times (𝐓_{, k} \times \vec{f} + (𝐓 \cdot {\hat{e}}_{i}) \otimes {\hat{e}}_{i} \times {\vec{f}}_{, k}) \\ = (\nabla \times 𝐓) \times \vec{f} - (𝐓 \cdot {\hat{e}}_{i}) \times {\hat{e}}_{k} \otimes {\hat{e}}_{i} \times {\vec{f}}_{, k} \\ = (\nabla \times 𝐓) \times \vec{f} - 𝐓 # (\nabla \otimes \vec{f}) \end{matrix}$

Beliebige Basis:

r o t (f^{i} {\vec{b}}_{i}) = g r a d (f^{i}) \times {\vec{b}}_{i} + f^{i} r o t ({\vec{b}}_{i})

Produkt mit Konstanten:

\begin{matrix} r o t (𝐂 \cdot \vec{f}) & = & - \vec{i} (𝐂 \cdot g r a d (\vec{f})) \\ \to r o t (\vec{f}) = r o t (𝟏 \cdot \vec{f}) = - \vec{i} (g r a d (\vec{f})) \\ r o t (\vec{f} \times 𝟏) & = & 𝟏 # g r a d (\vec{f})^{⊤} = d i v (\vec{f}) 𝟏 - g r a d (\vec{f}) \end{matrix}

In divergenzfreien Feldern ist also: $r o t (\vec{f} \times 𝟏) = - g r a d (\vec{f})$

Laplace-Operator

Definition des Laplace Operators/Allgemeines

Für Skalarfelder:^[4]

Δ := \nabla \cdot \nabla = \nabla^{2}

Δ = \sum_{i, j} g^{i j} \nabla_{i} \nabla_{j}

mit

g^{i j}

der metrische Tensor der Riemannschen Mannigfaltigkeit,

∇_i die kovariante Ableitung

Zusammenhang mit anderen Differentialoperatoren:

Δ f = d i v (g r a d (f)) = \nabla \cdot (\nabla f)

„Vektorieller Laplace-Operator“ in 3D:

Δ \vec{f} = d i v (g r a d (\vec{f})) = g r a d (d i v (\vec{f})) - r o t (r o t (\vec{f}))

Δ (\vec{x}) = \vec{0}

Für Tensorfelder^[1]

Δ (𝐓) \cdot \vec{c} = Δ (𝐓 \cdot \vec{c}) \forall \vec{c} \in 𝕍

Δ 𝐓 = g r a d (d i v (𝐓^{⊤}))^{⊤} - r o t (r o t (𝐓^{⊤})^{⊤})

Nützliche Formeln für den Laplace-Operator

\begin{matrix} Δ r = & \frac{2}{r} \\ Δ f (r) = & \frac{1}{r} \frac{d^{2}}{d r^{2}} (r f (r)) \\ Δ (r^{- 1}) = & 0 (r \neq 0) \\ Δ (f (r) \vec{x}) = & \vec{x} Δ f (r) + 2 g r a d f (r) \end{matrix}

mit $r = | \vec{x} |$ ^[2.1]

Laplace-Operator in verschiedenen Koordinatensystemen

\begin{matrix} Δ f = & f_{, k k} \\ Δ \vec{f} = & Δ f_{i} {\hat{e}}_{i} = f_{i, k k} {\hat{e}}_{i} \\ Δ 𝐓 = & Δ T_{i j} {\hat{e}}_{i} \otimes {\hat{e}}_{j} = T_{i j, k k} {\hat{e}}_{i} \otimes {\hat{e}}_{j} \end{matrix}

\begin{matrix} Δ f = & \frac{f_{, ρ}}{ρ} + f_{, ρ ρ} + \frac{f_{, φ φ}}{ρ^{2}} + f_{, z z} \\ Δ \vec{f} = & (Δ f_{ρ} - \frac{2 f_{φ, φ} + f_{ρ}}{ρ^{2}}) {\hat{e}}_{ρ} + (Δ f_{φ} + \frac{2 f_{ρ, φ} - f_{φ}}{ρ^{2}}) {\hat{e}}_{φ} + Δ f_{z} {\hat{e}}_{z} \end{matrix}