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 September 24th, 2018, 11:03 AM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Curvilinear Coordinates Volume Element Given r=(x(u,v,w), y(u,v,w), z(u,v,w)) How do you get from $\displaystyle dx=x_{u}du+x_{v}dv+x_{w}dw$ $\displaystyle dy=y_{u}du+y_{v}dv+y_{w}dw$ $\displaystyle dz= z_{u}du+z_{v}dv+z_{w}dw$ to $\displaystyle dxdydz=\biggr\vert\frac{\partial (x,y,z)}{\partial (u,v,w)}\biggr\vert dudvdw$ Look up Jacobian September 27th, 2018, 11:30 AM #2 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 $\displaystyle \vec{r}=x\vec{i}+y\vec{j}+z\vec{k}$ $\displaystyle d\vec{r}_{u}\equiv\frac{\partial \vec{r}}{\partial u}du=x_{u}du\vec{i}+y_{u}du\vec{j}+z_{u}du\vec{k}$ $\displaystyle d\vec{r}_{v}\equiv\frac{\partial \vec{r}}{\partial v}dv=x_{v}dv\vec{i}+y_{v}dv\vec{j}+z_{v}dv\vec{k}$ $\displaystyle d\vec{r}_{w}\equiv\frac{\partial \vec{r}}{\partial w}dw=x_{w}dw\vec{i}+y_{w}dw\vec{j}+z_{w}dw\vec{k}$ $\displaystyle d\vec{r}_{u}, d\vec{r}_{v}, d\vec{r}_{w}$ are distance vectors in x,y,z coordinate system and as such you can calculate the element of volume they form from $\displaystyle dV=|d\vec{r}_{u}\cdot d\vec{r}_{v}\times d\vec{r}_{w}| = Jdudvdw$ This is not dxdydz which is obvious if at any point you draw dx, dy, dz and $\displaystyle d\vec{r}_{u}, d\vec{r}_{v}, d\vec{r}_{w}$ However, $\displaystyle \int_{V}dxdydz=\int_{V}Jdudvdw$ which gives rise to the common fiction that dxdydz=Jdudvdw. dxdydz is given by: $\displaystyle dx=x_{u}du+x_{v}dv+x_{w}dw$ $\displaystyle dy=y_{u}du+y_{v}dv+y_{w}dw$ $\displaystyle dz=z_{u}du+z_{v}dv+z_{w}dw$ which in principle would give the same volume if you could do the integral. Tags coordinates, curvilinear, element, 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 Robotboyx9 Calculus 1 October 7th, 2017 10:39 PM Tyke Calculus 0 May 22nd, 2015 06:10 AM PaulScholes Linear Algebra 1 March 21st, 2014 06:42 PM mick7 Real Analysis 0 March 20th, 2014 09:16 AM Jhenrique Linear Algebra 0 February 19th, 2014 07:08 PM

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