vector differential calculus

Chapter 9: Vector Differential

Calculus

Topic 9.5: Curves & Arc Length1. 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)

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).

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.

Example 2: Ellipse (Parametric Representation)

Example 3: Straight Line (Parametric Representation)

Example 4: Circular Helix (Parametric Representation)

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

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

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

Related Problems from

Problem Set 9.5

Problems: 01 to 10

Problems: 11 to 18 Optional

Problems: 22 to 25

Problems: 26 to 28