these slides are related to vector differential calculus.

1. Chapter 9: Vector Differential
Calculus

2. Topic 9.5: Curves & Arc Length
1. Curves: Curves are of major applications of
differential calculus. (Another application is surfaces)
2. Any Curve C in space may occur as a path of a moving
body. That curve may be defined as parametric
representation i.e, function of a parameter t (time).
r (t) = [x (t), y (t), z (t)] = x (t) i + y (t) j + z (t) k
To each value t = to , there
corresponds a point of C
with position vector r(to)
with coordinates
x(to), y(to), z(to)

3. Types of Curves
 Plane Curve: A curve that lies in a plane in space (Circle in
Example 1).
 Twisted Curve: A curve that is not plane in space (Circular
Helix).
 Above two types of curves are also called simple curves
(curves without multiple points i.e, without points at which
the curve intersects or touches itself).

4. Example 1: Circle (Parametric Representation)
 Increasing time t is called the positive sense on C
defines direction of travel along C.
 Decreasing time t is called the negative sense on C
which defines direction of travel along C in
opposite direction.

5. Example 2: Ellipse (Parametric Representation)

6. Example 3: Straight Line (Parametric Representation)

7. Example 4: Circular Helix (Parametric Representation)

8. Tangent to a Curve (Tangent Vector)
 Tangent: Limiting position of straight line L touching a curve C
through two point P & Q as Q approaches P.
If C is given by r (t), P & Q corresponds to t &t+∆t,
then a vector in the direction of L is
 Tangent vector of C at point P is
 Unit Tangent Vector is given by

9. Tangent to a Curve (Tangent Line)
 Hence, the Tangent to Curve C at point P is
This is sum of position vector r of P and multiple
of tangent vector r’ of C at P.
 w is parameter here just like t.
 Compare this with simple line
equation r (t) = a + t b

10. Length of a Curve
 Length L of a curve C will be the limis of the lengths
of broken lines of n chords(n=5 in the fig) with larger
and larger n.
 approaches 0 as n∞
 Length L is given by
 Arc: Portion of a curve between any two point of it.
 Arc Length “s” of a curve C is given by

11. Related Problems from
Problem Set 9.5
Problems: 01 to 10
Problems: 11 to 18 Optional
Problems: 22 to 25
Problems: 26 to 28