See [[notes/Differentials#Differential Length Element|Differentials#Differential Length Element]]
See Coordinate Frame if you’re referring for robotics.
Cartesian coordinates are defined by 3 vectors with directions given by 𝑥̂, 𝑦̂, 𝑧̂.
In math, we use 𝑖̂, 𝑗̂, 𝑘̂. 𝐴⃗ in terms of the coordinate system will be given as:
where is the distance you move in 𝑥̂ direction, is the distance you move in 𝑦̂ direction and is the distance you move in 𝑧̂ direction. [[notes/Differentials#Differential Surface Elements|Differentials#Differential Surface Elements]]
We have two vectors which define the polar coordinates, vector 𝑟̂ which changes the radius of the compass and vector which changes the angle . is perpendicular to (and is thus, orthogonal) and by conventions changes the angle counter-clockwise. is called the azimuthal vector.
Intuition for Polar Coordinate
I get confused easily because there is an extra in for the differential area element, compared to cartesian coordinates which only has .
This is because the area, one of them is , and the arc is given by , NOT , is just an angle.
Cylindrical coordinates are basically polar coordinates in 3D.
We follow the left-hand rule and thus, 𝑟̂ is the first vector, ∅̂ is the second vector and 𝑧̂ is the third vector.
Conversion from Cartesian to Polar Coordinates
We simply let
We will make the z-axis as a principle axis of symmetry. So the sphere will be symmetrically created around the z-axis. We introduce a new vector, 𝜃̂ which changes the angle of the line with respect to the z-axis. This vector is called the polar vector.
Understand the 3 differential elements, should be easy now
In spherical coordinates, r is the distance of the point from the origin. In cylindrical coordinates, it is the perpendicular distance of the point from the z-axis.