Given a curve C1, the derivative is the curve C2 defined by the
derivation of the y component to the x component. So the y component of C2 gives the slope
When the curve is not smooth, or not a function, special effects can occur.
Suppose C1 is given by (x, y) = (f(t), g(t)) then C2 is given by y(x) = g '(x) / f '(x).
When the curve is formulated in polar coordinates, a polar pendant of the derivative is:
dr/dφ. This is then the slope in r as function of φ.
The reverse of differentiation is the operation of integration:
Given a curve C1, the integral is the curve C2 defined by the integration
of the y component to the x component. So the y component of C2 gives the cumulated area
between C1 and the x-as.
When C1 is given by (x, y) = (f(t), g(t)), then C2 has the form:
In polar coordinates, integration is over a sector between angles φ0 and φ.