Thus the curvature is the absolute value of the rate of change of with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at P and will be studied in greater detail in Chapter 11.

a) For a parametric curve x = x(t), y = y(t), derive the formula
where x' = dx/dt, y' = dy/dt. That is the primes denote differentiation with respect to t.

b) By regarding a curve y = f(x) as the parametric curve x = x, y = f(x), with parameter x, show that the formula in part (a) becomes

Solution: We are asked to find . From the graph we have:

By the definition of the arc length:

Differentiating the integral:

At the point we have:

In what follows:
y' = y'(t) = dy/dt, x' = x'(t) = dx/dt
We have to express the right side of (1) in terms of y' and x'. So:

That is:

Next:

Finally, from (1), (2) and (3):

where x' = dx/dt, y' = dy/dt.

This proves both parts of the problem.