- #1

Karl Karlsson1

- 2

- 0

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 \(\displaystyle \vec E_s\) and \(\displaystyle \vec E_t\) and the dual basis vectors \(\displaystyle \vec E^s\) and \(\displaystyle \vec E^t\)

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

My attempt:

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

Thanks in advance!