MHB Determine the area, calculate the basis vectors and determine the inner product

[Karl Karlsson1]

A coordinate system with the coordinates s and t in $$R^2$$ is defined by the coordinate transformations: $$ s = y/y_0$$ and $$t=y/y_0 - tan(x/x_0)$$ , where $$x_0$$ and $$y_0$$ 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 $$\vec E_s$$ and $$\vec E_t$$ and the dual basis vectors $$\vec E^s$$ and $$\vec E^t$$

c)Determine the inner products $$\vec E_s\cdot\vec E^s$$, $$\vec E_s\cdot\vec E^t$$, $$\vec E_t\cdot\vec E^s$$ and $$\vec E_t\cdot\vec E^t$$

My attempt:
a) Since $$tan(x/x_0)$$ is not defined for $$x=\pm\pi/2\cdot x_0$$ I assume x must be in between those values therefore $$-\pi/2\cdot x_0 < x < \pi/2\cdot x_0$$ 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 $$y=y_0\cdot s$$ and $$x=x_0\cdot arctan(s-t)$$. $$\vec E_s = \frac {x_0} {1 + (s-t)^2}\cdot\vec e-x + y_0\cdot\vec e_y$$ and $$\vec E_t = - \frac { x_0} { 1 + (s-t)^2}\cdot\vec e_x$$. I get the dual basis vectors from $$\vec E^s = \frac {1} {y_0}\cdot\vec e_y$$ and $$\vec E^t = \frac {1} {y_0}\cdot\vec e_y - \frac {1} {x_0(1+(x/x_0)^2)}\cdot\vec e_x$$ , 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 $$\vec E_s\cdot \vec E^s = 1 $$and$$\vec E_t\cdot \vec E^s = 0$$, this feels correct but then i get by just plugging in $$\vec E_t\cdot \vec E^t = \frac {x_0} {(1+(s-t)^2)(1+arctan(s-t)^2)} $$and $$\vec E_s\cdot \vec E^t = 1-\frac {1} {(1+(s-t)^2)(1+arctan(s-t)^2)}$$. Is this really correct? Because it feels like it is not correct.

Thanks in advance!

 

[Alone]

The tangent basis vectors are the ones given by: [math](\partial s/\partial x, \partial s/\partial y)_{s=s_0}=\vec{E}_s[/math], and the same with [math]E_t[/math] just replace t with s.
[math]s_0=s(x=0,y=0)[/math]
The area is given by https://www.euclideanspace.com/maths/algebra/vectors/vecGeometry/vec2d/index.htm

 

[soloenergy]

a) Your answer for the area is correct. The point (0,0) is well-defined for any value of y_0, as long as x is within the given range.

b) Your approach for calculating the tangent basis vectors and dual basis vectors is correct. However, there is a slight mistake in your calculation for \vec E^t. It should be \vec E^t = \frac{1}{y_0}\cdot\vec e_y - \frac{1}{x_0(1+(s-t)^2)}\cdot\vec e_x. This will give you the correct answer for part c).

c) Your answers for \vec E_s\cdot\vec E^s and \vec E_t\cdot\vec E^s are correct. However, the calculation for \vec E_t\cdot\vec E^t is incorrect. The correct answer is \vec E_t\cdot\vec E^t = \frac{x_0}{(1+(s-t)^2)(1+(x/x_0)^2)}. And for \vec E_s\cdot\vec E^t, the correct answer is \vec E_s\cdot\vec E^t = -\frac{1}{(1+(s-t)^2)(1+(x/x_0)^2)}.

Overall, your approach and calculations are correct, but there were some small mistakes in your calculations for \vec E^t and \vec E_t\cdot\vec E^t. Keep up the good work!