PX153 - A5 - coordinate systems

recall [basis vectors](PX153 - A1 - notation and geometrical representation)
- in 3D, a coordinate system can be defined by specifying 3 linearly independent basis vectors at every point in space
- usually defined to be orthonormal
to find $\underset{―}{{\hat{e}}_{i}}$ , several ways
- eg: tangent to the curve marking the change in coordinate

cartesian coordinates

2D
- here, $r$ is the magnitude of the vector and $θ$ is the angle made by the vector with the x-axis in counter-clockwise direction such that $0 \leq θ < 2 π$ or $θ \in [0, 2 π)$
transforming between polar and cartesian coordinates:
$x = r \cos θ$
$y = r \sin θ$
$r = \sqrt{x^{2} + y^{2}}$
$θ = \arctan (\frac{y}{x})$
the basis vectors for polar coordinates vary in space, but are always orthonormal
$\underset{―}{{\hat{e}}_{r}} = \cos θ \underset{―}{{\hat{e}}_{x}} + \sin θ \underset{―}{{\hat{e}}_{y}}$
$\underset{―}{{\hat{e}}_{θ}} = - \sin θ \underset{―}{{\hat{e}}_{x}} + \cos θ \underset{―}{{\hat{e}}_{y}}$
to check: these are orthonormal
${\vec{e}}_{r} \cdot {\vec{e}}_{θ} = \cos θ (- \sin θ) + \sin θ \cos θ = 0 ⟹ o r t h o r m a l$

3D
polar coordinates in x-y plane, extend by keeping the z-axis
- defined by $(r, θ, z)$
transformations to/from cartesian:
$x = r \cos θ$
$y = r \sin θ$
$z = z$
$r = \sqrt{x^{2} + y^{2}}$
$θ = \arctan \frac{y}{x}$
$z = z$