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* Page found: Normalschwingungen (eq math.1515.10)

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\begin{align}
  & {{v}_{x}}=\frac{dx}{dt}=\frac{\partial x}{\partial r}\dot{r}+\frac{\partial x}{\partial \vartheta }\dot{\vartheta }+\frac{\partial x}{\partial \phi }\dot{\phi }=\sin \vartheta \cos \phi \dot{r}+r\cos \vartheta \cos \phi \dot{\vartheta }-r\sin \vartheta \sin \phi \dot{\phi } \\
 & {{v}_{y}}=\frac{dy}{dt}=\frac{\partial y}{\partial r}\dot{r}+\frac{\partial y}{\partial \vartheta }\dot{\vartheta }+\frac{\partial y}{\partial \phi }\dot{\phi }=\sin \vartheta \sin \phi \dot{r}+r\cos \vartheta \sin \phi \dot{\vartheta }+r\sin \vartheta \cos \phi \dot{\phi } \\
 & {{v}_{z}}=\frac{dz}{dt}=\frac{\partial z}{\partial r}\dot{r}+\frac{\partial z}{\partial \vartheta }\dot{\vartheta }+\frac{\partial z}{\partial \phi }\dot{\phi }=\cos \vartheta \dot{r}-r\sin \vartheta \dot{\vartheta } \\
 &  \\
\end{align}

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v x = d ⁢ x d ⁢ t = ∂ ⁡ x ∂ ⁡ r ⁢ r ˙ + ∂ ⁡ x ∂ ⁡ ϑ ⁢ ϑ ˙ + ∂ ⁡ x ∂ ⁡ ϕ ⁢ ϕ ˙ = sin ⁡ ϑ ⁢ cos ⁡ ϕ ⁢ r ˙ + r ⁢ cos ⁡ ϑ ⁢ cos ⁡ ϕ ⁢ ϑ ˙ - r ⁢ sin ⁡ ϑ ⁢ sin ⁡ ϕ ⁢ ϕ ˙ v y = d ⁢ y d ⁢ t = ∂ ⁡ y ∂ ⁡ r ⁢ r ˙ + ∂ ⁡ y ∂ ⁡ ϑ ⁢ ϑ ˙ + ∂ ⁡ y ∂ ⁡ ϕ ⁢ ϕ ˙ = sin ⁡ ϑ ⁢ sin ⁡ ϕ ⁢ r ˙ + r ⁢ cos ⁡ ϑ ⁢ sin ⁡ ϕ ⁢ ϑ ˙ + r ⁢ sin ⁡ ϑ ⁢ cos ⁡ ϕ ⁢ ϕ ˙ v z = d ⁢ z d ⁢ t = ∂ ⁡ z ∂ ⁡ r ⁢ r ˙ + ∂ ⁡ z ∂ ⁡ ϑ ⁢ ϑ ˙ + ∂ ⁡ z ∂ ⁡ ϕ ⁢ ϕ ˙ = cos ⁡ ϑ ⁢ r ˙ - r ⁢ sin ⁡ ϑ ⁢ ϑ ˙ missing-subexpression subscript 𝑣 𝑥 𝑑 𝑥 𝑑 𝑡 𝑥 𝑟 ˙ 𝑟 𝑥 italic-ϑ ˙ italic-ϑ 𝑥 italic-ϕ ˙ italic-ϕ italic-ϑ italic-ϕ ˙ 𝑟 𝑟 italic-ϑ italic-ϕ ˙ italic-ϑ 𝑟 italic-ϑ italic-ϕ ˙ italic-ϕ missing-subexpression subscript 𝑣 𝑦 𝑑 𝑦 𝑑 𝑡 𝑦 𝑟 ˙ 𝑟 𝑦 italic-ϑ ˙ italic-ϑ 𝑦 italic-ϕ ˙ italic-ϕ italic-ϑ italic-ϕ ˙ 𝑟 𝑟 italic-ϑ italic-ϕ ˙ italic-ϑ 𝑟 italic-ϑ italic-ϕ ˙ italic-ϕ missing-subexpression subscript 𝑣 𝑧 𝑑 𝑧 𝑑 𝑡 𝑧 𝑟 ˙ 𝑟 𝑧 italic-ϑ ˙ italic-ϑ 𝑧 italic-ϕ ˙ italic-ϕ italic-ϑ ˙ 𝑟 𝑟 italic-ϑ ˙ italic-ϑ missing-subexpression missing-subexpression {\displaystyle{\displaystyle\begin{aligned} &\displaystyle{{v}_{x}}=\frac{dx}{% dt}=\frac{\partial x}{\partial r}\dot{r}+\frac{\partial x}{\partial\vartheta}% \dot{\vartheta}+\frac{\partial x}{\partial\phi}\dot{\phi}=\sin\vartheta\cos% \phi\dot{r}+r\cos\vartheta\cos\phi\dot{\vartheta}-r\sin\vartheta\sin\phi\dot{% \phi}\\ &\displaystyle{{v}_{y}}=\frac{dy}{dt}=\frac{\partial y}{\partial r}\dot{r}+% \frac{\partial y}{\partial\vartheta}\dot{\vartheta}+\frac{\partial y}{\partial% \phi}\dot{\phi}=\sin\vartheta\sin\phi\dot{r}+r\cos\vartheta\sin\phi\dot{% \vartheta}+r\sin\vartheta\cos\phi\dot{\phi}\\ &\displaystyle{{v}_{z}}=\frac{dz}{dt}=\frac{\partial z}{\partial r}\dot{r}+% \frac{\partial z}{\partial\vartheta}\dot{\vartheta}+\frac{\partial z}{\partial% \phi}\dot{\phi}=\cos\vartheta\dot{r}-r\sin\vartheta\dot{\vartheta}\\ &\\ \end{aligned}}}

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vx=dxdt=∂x∂rr˙+∂x∂ϑϑ˙+∂x∂ϕϕ˙=sin⁡ϑcos⁡ϕr˙+rcos⁡ϑcos⁡ϕϑ˙−rsin⁡ϑsin⁡ϕϕ˙vy=dydt=∂y∂rr˙+∂y∂ϑϑ˙+∂y∂ϕϕ˙=sin⁡ϑsin⁡ϕr˙+rcos⁡ϑsin⁡ϕϑ˙+rsin⁡ϑcos⁡ϕϕ˙vz=dzdt=∂z∂rr˙+∂z∂ϑϑ˙+∂z∂ϕϕ˙=cos⁡ϑr˙−rsin⁡ϑϑ˙

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