Coordinate System

This was focused on heavily for ECE106, and also learned in MATH119.

See [[notes/Differentials#Differential Length Element|Differentials#Differential Length Element]]

See Coordinate Frame if you’re referring for robotics.

Cartesian Coordinate

Cartesian coordinates are defined by 3 vectors with directions given by π‘₯Μ‚, 𝑦̂, 𝑧̂.

In math, we use 𝑖̂, 𝑗̂, π‘˜Μ‚. 𝐴⃗ in terms of the coordinate system will be given as:

where is the distance you move in π‘₯Μ‚ 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]]

Polar Coordinates

We have two vectors which define the polar coordinates, vector π‘ŸΜ‚ 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 Coordinate

Cylindrical coordinates are basically polar coordinates in 3D.

We follow the left-hand rule and thus, π‘ŸΜ‚ is the first vector, βˆ…Μ‚ is the second vector and 𝑧̂ is the third vector.

Conversion from Cartesian to Polar Coordinates

We simply let

Spherical Coordinate

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, πœƒΜ‚ 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.