Polar Coordinates

Arc Length
Area
Polar Coordinates

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Introduction

We can describe the location of a point by giving its distance from the origin and the angle from the horizontal axis .
, but can take on any value. This means polar representations are not unique.

Notation

In physics, the notation and (or ) is used, whereas in most mathematical text and are used.

In fact, the use of and can make the understanding of polar coordinates easier.

We can describe our and with sine and cosine:

We can also describe:

Example

The circle can be described as . Now consider the equation , with the restriction that . What does the graph look like?

Important

For MATH 117 at least, when , since will produce a negative value, we also reflect the coordinates on the axis, which produces the same circle as above.

Example

Consider . First, note that when , or when and .
Polar Coordinates Ex 2.png

The Jacobian for Cartesian to Polar

See Jacobian

Definition

With

(see determinant, Partial Derivatives, Pythagorean identities)

Equation