Formelsammlung Tensoranalysis

grad (f) = f_{, i} {\hat{e}}_{i}

grad (\vec{f}) = {\vec{f}}_{, i} \otimes {\hat{e}}_{i} = {\hat{e}}_{i} \otimes grad (f_{i}) = f_{i, j} {\hat{e}}_{i} \otimes {\hat{e}}_{j}

grad (f) = f_{, ρ} {\hat{e}}_{ρ} + \frac{f_{, φ}}{ρ} {\hat{e}}_{φ} + f_{, z} {\hat{e}}_{z}

\begin{aligned} grad (\vec{f}) = & {\hat{e}}_{ρ} \otimes grad (f_{ρ}) + {\hat{e}}_{φ} \otimes grad (f_{φ}) + {\hat{e}}_{z} \otimes grad (f_{z}) \\ + \frac{1}{ρ} (f_{ρ} {\hat{e}}_{φ} - f_{φ} {\hat{e}}_{ρ}) \otimes {\hat{e}}_{φ} \end{aligned}

#Krummlinige Koordinaten:

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

\begin{aligned} grad (\vec{f}) = & {\hat{e}}_{r} \otimes grad (f_{r}) + {\hat{e}}_{ϑ} \otimes grad (f_{ϑ}) + {\hat{e}}_{φ} \otimes grad (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{aligned}

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

Vektorfelder:

grad ({\vec{g}}_{i}) = Γ_{i j}^{k} {\vec{g}}_{k} \otimes {\vec{g}}^{j}

grad ({\vec{g}}^{k}) = - Γ_{i j}^{k} {\vec{g}}^{i} \otimes {\vec{g}}^{j}

grad (f^{i} {\vec{g}}_{i}) = {f^{i} |}_{j} {\vec{g}}_{i} \otimes {\vec{g}}^{j}

grad (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:

grad (T) [\vec{h}] = (\vec{h} \cdot {\vec{g}}^{k}) T_{, k} = \vec{h} \cdot ({\vec{g}}^{k} \otimes T_{, k}) = (T_{, k} \otimes {\vec{g}}^{k}) \cdot \vec{h}

Soll das Argument wie beim Vektorgradient rechts vom Operator stehen, dann lautet der Tensorgradient

grad (T) = T_{, k} \otimes {\vec{g}}^{k}

Für ein Tensorfeld zweiter Stufe:

\begin{aligned} grad (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} \\ grad (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} \\ grad (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} \\ grad (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{aligned}

Produktregel für Gradienten

\begin{array}{rclcl} grad (f g) & = & (f_{, i} g + f g_{, i}) {\hat{e}}_{i} & = & grad (f) g + f grad (g) \\ grad (f \vec{g}) & = & (f_{, i} \vec{g} + f {\vec{g}}_{, i}) \otimes {\hat{e}}_{i} & = & \vec{g} \otimes grad (f) + f grad (\vec{g}) \\ grad (\vec{f} \cdot \vec{g}) & = & ({\vec{f}}_{, i} \cdot \vec{g} + \vec{f} \cdot {\vec{g}}_{, i}) {\hat{e}}_{i} & = & \vec{g} \cdot grad (\vec{f}) + \vec{f} \cdot grad (\vec{g}) \\ grad (\vec{f} \times \vec{g}) & = & ({\vec{f}}_{, i} \times \vec{g} + \vec{f} \times {\vec{g}}_{, i}) \otimes {\hat{e}}_{i} & = & \vec{f} \times grad (\vec{g}) - \vec{g} \times grad (\vec{f}) \end{array}

In drei Dimensionen ist speziell^[3]

grad (\vec{f} \cdot \vec{g}) = grad (\vec{f}) \cdot \vec{g} + grad (\vec{g}) \cdot \vec{f} + \vec{f} \times rot (\vec{g}) + \vec{g} \times rot (\vec{f})

Beliebige Basis:

grad (f_{i} {\vec{b}}_{i}) = {\vec{b}}_{i} \otimes grad (f_{i}) + f_{i} grad ({\vec{b}}_{i})

Divergenz

Definition der Divergenz/Allgemeines

Vektorfeld $\vec{f}$ :

div (\vec{f}) = \nabla \cdot \vec{f} = Sp (grad (\vec{f}))

div (\vec{x}) = Sp (grad (\vec{x})) = Sp (1) = 3

Klassische Definition für ein Tensorfeld T:^[1]

div (T) \cdot \vec{c} = div (T^{⊤} \cdot \vec{c}) \forall \vec{c} \in V

→

div (T) = \nabla \cdot (T^{⊤})

Koordinatenfreie Darstellung:

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

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

Zusammenhang mit den anderen Differentialoperatoren:

\begin{array}{lcccl} div (\vec{f}) & = & \nabla \cdot \vec{f} & = & Sp (grad (\vec{f})) \\ div (f 1) & = & \nabla \cdot (f 1) & = & grad (f) \end{array}

Divergenz in verschiedenen Koordinatensystemen

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

div (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}

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

\begin{aligned} div (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{aligned}

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

\begin{aligned} div (\vec{f}) = & f_{r, r} + \frac{2 f_{r} + f_{ϑ, ϑ}}{r} + \frac{f_{ϑ} \cos (ϑ) + f_{φ, φ}}{r \sin (ϑ)} \\ div (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{aligned}

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

Produktregel für Divergenzen

div (f \vec{g}) = \nabla \cdot (f \vec{g}) = (f_{, i} \vec{g} + f {\vec{g}}_{, i}) \cdot {\hat{e}}_{i} = grad (f) \cdot \vec{g} + f div (\vec{g})

div (\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 rot (\vec{f}) - \vec{f} \cdot rot (\vec{g})

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

\begin{aligned} \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{aligned}

Beliebige Basis:

div (f_{i} {\vec{b}}_{i}) = \nabla \cdot (f_{i} {\vec{b}}_{i}) = grad (f_{i}) \cdot {\vec{b}}_{i} + f_{i} div ({\vec{b}}_{i})

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

div (f C) = C \cdot grad (f) \to div (f 1) = grad (f)

\nabla \cdot (f C) = grad (f) \cdot C \to \nabla \cdot (f 1) = \nabla f

\begin{aligned} div (C \cdot \vec{f}) = C^{⊤} : grad (\vec{f}) \to div (\vec{f}) = & div (1 \cdot \vec{f}) = 1 : grad (\vec{f}) \\ = & Sp (grad (\vec{f})) \end{aligned}

Rotation

Definition der Rotation/Allgemeines

Vektorfeld $\vec{f}$ :

rot (\vec{f}) = \nabla \times \vec{f}

Klassische Definition für ein Tensorfeld T:^[1]

rot (T) \cdot \vec{c} = rot (T^{⊤} \cdot \vec{c}) \forall \vec{c} \in V

→

rot (T) = \nabla \times (T^{⊤})

Allgemeine Identitäten:

{T = T}^{⊤} \to Sp (rot (T)) = Sp (\nabla \times T) = 0

rot (\vec{x}) = \vec{0}

Integrabilitätsbedingung^[4]: Jedes divergenzfreie Vektorfeld ist die Rotation eines Vektorfeldes:

div (\vec{f}) = 0 \to \exists \vec{g} : \vec{f} = rot (\vec{g})

.

Siehe auch #Satz über rotationsfreie Felder.

Koordinatenfreie Darstellung:

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

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

Zusammenhang mit den anderen Differentialoperatoren:

\begin{aligned} rot (f \vec{c}) = & grad (f) \times \vec{c} \\ rot (\vec{f}) = & - \vec{i} (grad (\vec{f})) = \vec{i} (\nabla \otimes \vec{f}) = \nabla \times \vec{f} \end{aligned}

Rotation in verschiedenen Koordinatensystemen

\begin{aligned} rot (\vec{f}) = & 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{aligned}

rot (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}

rot (\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}

rot (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{aligned} rot (\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{aligned}

rot (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{aligned} rot (f \vec{g}) = & {\hat{e}}_{i} \times (f_{, i} \vec{g} + f {\vec{g}}_{, i}) = grad (f) \times \vec{g} + f rot (\vec{g}) \\ rot (\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} \\ = & grad (\vec{f}) \cdot \vec{g} - div (\vec{f}) \vec{g} + div (\vec{g}) \vec{f} - grad (\vec{g}) \cdot \vec{f} \\ = & div (\vec{f} \otimes \vec{g}) - div (\vec{g} \otimes \vec{f}) = \nabla \cdot (\vec{g} \otimes \vec{f}) - \nabla \cdot (\vec{f} \otimes \vec{g}) \end{aligned}

\begin{aligned} rot (\vec{f} \otimes \vec{g}) = & {\hat{e}}_{i} \times ({\vec{g}}_{, i} \otimes \vec{f} + \vec{g} \otimes {\vec{f}}_{, i}) & = & rot (\vec{g}) \otimes \vec{f} - \vec{g} \times grad (\vec{f})^{⊤} \\ rot (f T) = & {\hat{e}}_{k} \times (f_{, k} T^{⊤} + f T_{, k}^{⊤}) & = & grad (f) \times (T^{⊤}) + f rot (T) \end{aligned}

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

\begin{aligned} \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{aligned}

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

Beliebige Basis:

rot (f^{i} {\vec{b}}_{i}) = grad (f^{i}) \times {\vec{b}}_{i} + f^{i} rot ({\vec{b}}_{i})

Produkt mit Konstanten:

\begin{array}{rcl} rot (C \cdot \vec{f}) & = & - \vec{i} (C \cdot grad (\vec{f})) \\ \to rot (\vec{f}) = rot (1 \cdot \vec{f}) = - \vec{i} (grad (\vec{f})) \\ rot (\vec{f} \times 1) & = & 1 # grad (\vec{f})^{⊤} = grad (\vec{f}) - div (\vec{f}) 1 \end{array}

In divergenzfreien Feldern ist also: $rot (\vec{f} \times 1) = grad (\vec{f})$

Laplace-Operator

Definition/Allgemeines

Δ := \nabla \cdot \nabla = \nabla^{2}

Zusammenhang mit anderen Differentialoperatoren:

\begin{array}{rclcl} Δ f & = & div (grad (f)) & = & \nabla \cdot (\nabla f) \\ Δ \vec{f} & = & div (grad (\vec{f})) & = & \nabla \cdot (\nabla \otimes \vec{f}) \end{array}

„Vektorieller Laplace-Operator“:

Δ \vec{f} = grad (div (\vec{f})) - rot (rot (\vec{f}))

Laplace-Operator in verschiedenen Koordinatensystemen

\begin{aligned} 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{aligned}

\begin{aligned} 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{aligned}