Karl Karlsson
New member
- Joined
- Nov 4, 2019
- Messages
- 9
A coordinate system with the coordinates s and t in ##R^2## is defined by the coordinate transformations: [MATH] s = y/y_0[/MATH] and [MATH]t=y/y_0 - tan(x/x_0)[/MATH] , where [MATH]x_0[/MATH] and [MATH]y_0[/MATH] are constants.
a) Determine the area that includes the point (x, y) = (0, 0) where the coordinate system
is well defined. Express the area both in the Cartesian coordinates (x, y) and in
the new coordinates (s, t).
b) Calculate the tangent basis vectors [MATH]\vec E_s[/MATH] and [MATH]\vec E_t[/MATH] and the dual basis vectors [MATH]\vec E^s[/MATH] and [MATH]\vec E^t[/MATH]
c)Determine the inner products [MATH]\vec E_s\cdot\vec E^s[/MATH], [MATH]\vec E_s\cdot\vec E^t[/MATH], [MATH]\vec E_t\cdot\vec E^s[/MATH] and [MATH]\vec E_t\cdot\vec E^t[/MATH]
My attempt:
a) Since [MATH]tan(x/x_0)[/MATH] is not defined for [MATH]x=\pm\pi/2\cdot x_0[/MATH] I assume x must be in between those values therefore [MATH]-\pi/2\cdot x_0 < x < \pi/2\cdot x_0[/MATH] and y can be any real number. Is this the correct answer on a)?
b) I can solve x and y for s and t which gives me [MATH]y=y_0\cdot s[/MATH] and [MATH]x=x_0\cdot arctan(s-t)[/MATH]. [MATH]\vec E_s = \frac {x_0} {1 + (s-t)^2}\cdot\vec e-x + y_0\cdot\vec e_y[/MATH] and [MATH]\vec E_t = - \frac { x_0} { 1 + (s-t)^2}\cdot\vec e_x[/MATH]. I get the dual basis vectors from [MATH]\vec E^s = \frac {1} {y_0}\cdot\vec e_y[/MATH] and [MATH]\vec E^t = \frac {1} {y_0}\cdot\vec e_y - \frac {1} {x_0(1+(x/x_0)^2)}\cdot\vec e_x[/MATH] , is this the correct approach?
c) It was here that I really started to question if i had done correct on a and b since I get [MATH]\vec E_s\cdot \vec E^s = 1 [/MATH]and[MATH]\vec E_t\cdot \vec E^s = 0[/MATH], this feels correct but then i get by just plugging in [MATH]\vec E_t\cdot \vec E^t = \frac {x_0} {(1+(s-t)^2)(1+arctan(s-t)^2)} [/MATH]and [MATH]\vec E_s\cdot \vec E^t = 1-\frac {1} {(1+(s-t)^2)(1+arctan(s-t)^2)}[/MATH]. Is this really correct? Because it feels like it is not correct.
Thanks in advance!
a) Determine the area that includes the point (x, y) = (0, 0) where the coordinate system
is well defined. Express the area both in the Cartesian coordinates (x, y) and in
the new coordinates (s, t).
b) Calculate the tangent basis vectors [MATH]\vec E_s[/MATH] and [MATH]\vec E_t[/MATH] and the dual basis vectors [MATH]\vec E^s[/MATH] and [MATH]\vec E^t[/MATH]
c)Determine the inner products [MATH]\vec E_s\cdot\vec E^s[/MATH], [MATH]\vec E_s\cdot\vec E^t[/MATH], [MATH]\vec E_t\cdot\vec E^s[/MATH] and [MATH]\vec E_t\cdot\vec E^t[/MATH]
My attempt:
a) Since [MATH]tan(x/x_0)[/MATH] is not defined for [MATH]x=\pm\pi/2\cdot x_0[/MATH] I assume x must be in between those values therefore [MATH]-\pi/2\cdot x_0 < x < \pi/2\cdot x_0[/MATH] and y can be any real number. Is this the correct answer on a)?
b) I can solve x and y for s and t which gives me [MATH]y=y_0\cdot s[/MATH] and [MATH]x=x_0\cdot arctan(s-t)[/MATH]. [MATH]\vec E_s = \frac {x_0} {1 + (s-t)^2}\cdot\vec e-x + y_0\cdot\vec e_y[/MATH] and [MATH]\vec E_t = - \frac { x_0} { 1 + (s-t)^2}\cdot\vec e_x[/MATH]. I get the dual basis vectors from [MATH]\vec E^s = \frac {1} {y_0}\cdot\vec e_y[/MATH] and [MATH]\vec E^t = \frac {1} {y_0}\cdot\vec e_y - \frac {1} {x_0(1+(x/x_0)^2)}\cdot\vec e_x[/MATH] , is this the correct approach?
c) It was here that I really started to question if i had done correct on a and b since I get [MATH]\vec E_s\cdot \vec E^s = 1 [/MATH]and[MATH]\vec E_t\cdot \vec E^s = 0[/MATH], this feels correct but then i get by just plugging in [MATH]\vec E_t\cdot \vec E^t = \frac {x_0} {(1+(s-t)^2)(1+arctan(s-t)^2)} [/MATH]and [MATH]\vec E_s\cdot \vec E^t = 1-\frac {1} {(1+(s-t)^2)(1+arctan(s-t)^2)}[/MATH]. Is this really correct? Because it feels like it is not correct.
Thanks in advance!