Mathematical Tools for Physics

1—Basic Stuff 20

1.7 Polar Coordinates
When you compute an integral in the plane, you need the element of area appropriate to the coordinate system
that you’re using. In the most common case, that of rectangular coordinates, you find the element of area by
drawing the two lines at constant coordinatesxandx+dx. Then you draw the two lines at constant coordinates
yandy+dy. The little rectangle that they circumscribe has an areadA=dxdy.

x x+dx

y

y+dy

r r+dr

θ

θ+dθ

In polar coordinates you do exactly the same thing! The coordinates arerandθ, and the line at constant
radiusrand at constantr+drdefine two neighboring circles. The lines at constant angleθand at constant
angleθ+dθform two closely spaced rays from the origin. These four lines circumscribe a tiny area that is, for
small enoughdranddθ, a rectangle. You then know its area is the product of its two sides*:dA= (dr)(r dθ).
This is the basic element of area for polar coordinates.
The area of a circle is the sum of all the pieces of area within it
∫
dA=

∫R

0

r dr

∫ 2 π

0

dθ

I find it most useful to write double integrals in this way, so that the limits of integration are right next to
the differential. The other notation can put the differential a long distance from where you show the limits of
integration. I get less confused my way.

∫R

0

r dr

∫ 2 π

0

dθ=

∫R

0

r dr 2 π= 2πR^2 /2 =πR^2

• If you’re tempted to say that the area isdA=dr dθ,look at the dimensions.This expression is a length,
  not an area.