Polar coordinates - multivariable calculus

Polar coordinates

As an alternative to Cartesian coordinates using distances along the - and -axis respectively, we can use polar coordinates to represent a point in two dimensions.

This form uses only one distance from the origin, as well as an angle , measured in the counter-clockwise direction from the positive -axis. Hence, a point in polar form is written as .

The coordinate transformation from Cartesian to polar coordinates looks as follows:

Cylindrical coordinates

If we equip the polar coordinate system with a -axis, we get a coordinate system in 3 dimensions called cylindrical coordinates.

Cylindrical coordinates are polar coordinates with a -axis

As per usual, we use our right hand to define the directional relationships in the coordinate system:

The right hand rule for direction of rotation:

Using the right hand, let the thumb point in the direction of the -axis, and curl your fingers.

Then, if you twist the hand in the direction that the fingers point, this will define the direction of rotation for the angle .

To visualize what cylindrical coordinates can look like in practice, imagine a car driving up a circular ramp in a parking garage. The position of the car, taken from the center of the ramp as the origin, can be described using the radius of the ramp, the current angle to some defined horizontal direction (the regular -axis), and the height given by the -axis.

This example highlights one of the pros of cylindrical coordinates. Since the radius remains unchanged during the drive up the ramp, the only changing quantities in this particular case are the angle and the height.

A point expressed in cylindrical coordinates takes the form , and the coordinate transformation from the Cartesian coordinate system in 3 dimensions look as follows:

From these formulas, we can derive a set of equations to convert coordinates the other way around too:

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