Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Successfully reported this slideshow.

Like this presentation? Why not share!

987 views

Published on

these slides are related to vector differential calculus.

Published in:
Education

No Downloads

Total views

987

On SlideShare

0

From Embeds

0

Number of Embeds

6

Shares

0

Downloads

26

Comments

0

Likes

1

No embeds

No notes for slide

- 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

No public clipboards found for this slide

Be the first to comment