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 April 18th, 2015, 08:46 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 09:38 AM. Tags field, induced, moving, surface, vector, volume Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Jhenrique Calculus 2 January 1st, 2014 04:08 PM FishFace Physics 3 October 27th, 2011 06:39 AM MasterOfDisaster Calculus 2 September 26th, 2011 09:17 AM FKeeL Computer Science 0 September 10th, 2009 06:58 AM mato Linear Algebra 0 December 5th, 2007 03:05 PM

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