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April 18th, 2015, 09:46 AM  #1 
Newbie Joined: Apr 2015 From: Finland Posts: 2 Thanks: 0  Volume induced by a vector field across a moving surface
We are given a vector field $\mathbf{F} \colon \mathbb{R}^3 \to \mathbb{R}^3$ and a parametric surface $\mathcal{S}$ defined by $\mathbf{r} \colon A \to \mathbb{R}^3$, where $A \subset \mathbb{R}^2$. Now the flux induced by $\mathbf{F}$ through $\mathbf{r}$ is \begin{align} \iint_{\mathcal{S}} \mathbf{F} \cdot \hat{\mathbf{N}} \,\mathrm{d}S &= \iint_A \Big\langle \mathbf{F}(\mathbf{r}(u, v)), \frac{\partial \mathbf{r}} {\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \Big\rangle \, \mathrm{d}u \, \mathrm{d}v. \end{align} Now suppose that the vector field changes with time, i.e., we have $\mathbf{F} \colon \mathbb{R}^4 \to \mathbb{R}^3$, and the parametric surface $\mathcal{S}$ evolves in time as well: $\mathbf{r} \colon A \times \mathbb{R} \to \mathbb{R}^3$. If $\hat{\mathbf{N}}$ is the unit normal to surface $\mathcal{S}$, it appears to me that the area element "passes" space with rate \begin{align} \Big\langle \hat{\mathbf{N}}, \frac{ \partial \mathbf{r} }{ \partial t} \Big\rangle, \end{align} and so everything boils down to \begin{align} \Phi(t) &= \iint_{\mathcal{S}} \langle F, \hat{\mathbf{N}} \rangle  \Big\langle \frac{\partial \mathbf{r}}{\partial t}, \hat{\mathbf{N}} \Big\rangle \, \mathrm{d}S\\ &= \iint_{\mathcal{S}} \Big\langle \mathbf{F}  \frac{\partial \mathbf{r}}{\partial t} , \hat{\mathbf{N}} \Big\rangle \, \mathrm{d}S \\ &= \iint_A \Big\langle \mathbf{F}(\mathbf{r}(u, v, t), t)  \frac{\partial \mathbf{r}}{\partial t}, \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v} \Big\rangle \, \mathrm{d}u \, \mathrm{d}v. \end{align} Now the amount of volume flowing according to the vector field $\mathbf{F}$ through the moving surface $\mathcal{S}$ during time interval $[t_a, t_b]$ is simply \begin{align} V = \int_{t_a}^{t_b} \Phi(t) \, \mathrm{d}t. \end{align} Is this correct? Last edited by coderodde; April 18th, 2015 at 10:38 AM. 

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field, induced, moving, surface, vector, volume 
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