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Dated: 30-10-2024

Change of Parameter

\[\vec r(t) = 3 \cos (t) \hat i + 3 \sin(t) \hat j \text{ where } 0 \le t \le 2 \pi\]

This is the parametric equation for a circle with \(\text{radius} = 3\), where \(\vec r\) depends on the angle.

\[\because l = r \theta\]
\[\frac l r = \theta\]
\[\therefore t = \frac l 3\]
\[0 \times 3 \le l \le 2 \pi \times 3\]
\[0 \le l \le 6 \pi\]

Hence, our equation becomes

\[\vec r(l) = 3 \cos \left(\frac l 3\right) \hat i + 3 \sin \left( \frac l 3\right) \hat j \text{ where } 0 \le l \le 6 \pi\]

\(t = g(u)\) is called a smooth change in parameter if \(g(u)\) satisfies the following conditions::

  1. \(g\) is differentiable1
  2. \(g^\prime\) is continuous2
  3. \(g^\prime(u) \ne 0\) for any \(u\) in the domain3 of \(g\)

Arc Length

Because we do derivative2 of vector valued functions4 component wise, we can also do integration5 component wise.

Example

if \(x^\prime (t)\) and \(y^\prime (t)\) are continuous2 for \(a \le t \le b\) then the arc length is

\[L = \int_a^b \sqrt{\left(\frac {dx}{dt}\right)^2 + \left(\frac {dy}{dt}\right)^2} dt\]

Arc Length as Parameter

  1. Select an arbitrary point P on the curve C to serve as a reference point.
  2. Choose one direction to move along the curve which will be considered positive and other direction will be considered negative.

References

Read more about notations and symbols.

  1. Read more about differentiation. ↩

  2. Read more about continuity. ↩↩↩

  3. Read more about functions. ↩

  4. Read more about vector valued functions. ↩

  5. Read more about integrals. ↩