Win_odd Dhamnekar
Junior Member
- Joined
- Aug 14, 2018
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How to sketch several coordinate curves of the given coordinate system to form a grid of "rectangles" (i.e., make sure the u-curves are close enough to appear straight between the v-curves and vice-versa. How to find the area differential and discuss its relationship to the "coordinate curve grid". The following are the linear transformations and have a constant jacobian determinant.
1)Elliptic coordinates \(\displaystyle T(u,v)=\langle \cosh{u} \cos{v},\sinh{u} sin{v}\rangle\)
2)Bipolar coordinates \(\displaystyle T(u,v)=\langle \frac{\sinh{v}}{\cosh{v} \cos{u}},\frac {\sin{u}}{\cosh{v}\cos{u}}\rangle\)
Answer : 1)The area differential of elliptic coordinates is \(\displaystyle (\sinh^2{u}cos^2{v} + \cosh^2{u}\sin^2{v})dudv\) But how to sketch coordinate curves of the given cordinate system?
2) The area differential of bipolar coordinates is \(\displaystyle \bigg(\frac{\sinh^2{v}\sin^2{u}}{(\cosh{v}-\cos{u})^4}-\bigg((\frac{\cosh{v}}{(\cosh{v}-\cos{u})}-\frac{\sin^2{v}}{(\cosh{v}-\cos{u})^2}) (\frac{\cos{u}}{\cosh{v}-cos{u}}-\frac{\sin^2{u}}{(\cosh{v}-\cos{u})^2})\bigg)\bigg) du dv\)
How to sketch coordinte curves of the given coordinate system?
1)Elliptic coordinates \(\displaystyle T(u,v)=\langle \cosh{u} \cos{v},\sinh{u} sin{v}\rangle\)
2)Bipolar coordinates \(\displaystyle T(u,v)=\langle \frac{\sinh{v}}{\cosh{v} \cos{u}},\frac {\sin{u}}{\cosh{v}\cos{u}}\rangle\)
Answer : 1)The area differential of elliptic coordinates is \(\displaystyle (\sinh^2{u}cos^2{v} + \cosh^2{u}\sin^2{v})dudv\) But how to sketch coordinate curves of the given cordinate system?
2) The area differential of bipolar coordinates is \(\displaystyle \bigg(\frac{\sinh^2{v}\sin^2{u}}{(\cosh{v}-\cos{u})^4}-\bigg((\frac{\cosh{v}}{(\cosh{v}-\cos{u})}-\frac{\sin^2{v}}{(\cosh{v}-\cos{u})^2}) (\frac{\cos{u}}{\cosh{v}-cos{u}}-\frac{\sin^2{u}}{(\cosh{v}-\cos{u})^2})\bigg)\bigg) du dv\)
How to sketch coordinte curves of the given coordinate system?
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