Polar Coordinates

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Section 1.1 Polar Coordinates

Polar coordinates are useful for situations with circular symmetry in the plane. The polar coordinates (\(r\text{,}\)\(\phi\)) of a point \(P\) are given by the distance \(r\) of \(P\) from the origin and the angle \(\phi\) from the positive \(x\)-axis to \(P\text{,}\) as shown in Figure 1.1.

Figure 1.1. The construction of the polar coordinates (\(r\text{,}\)\(\phi\)) at an arbitrary point.

Notation: When we think of the plane as a cross-section of spherical coordinates, we will use the pair (\(r\text{,}\)\(\phi\)) for polar coordinates. When we think of the plane as a cross-section of cylindricals coordinates, we will use the pair (\(s\text{,}\)\(\phi\)) for polar coordinates. In other references, you may also see the angle called \(\theta\) instead of \(\phi\text{;}\) we use \(\phi\) to agree with our conventions for (cylindrical and) spherical coordinates.

It is important to remember that the angle \(\phi\) does not measure distance; it has the wrong dimensions. Angles (in radians) are defined as the ratio of arclength to radius on the circle, so the arclength from the positive \(x\)-axis to \(P\) along the circle shown in the figure is \(r\phi\text{.}\)