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# Vector differential Calculus

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### Vector differential Calculus

1. 1. Chapter 9: Vector Differential Calculus
2. 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. 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. 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. 5. Example 2: Ellipse (Parametric Representation)
6. 6. Example 3: Straight Line (Parametric Representation)
7. 7. Example 4: Circular Helix (Parametric Representation)
8. 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. 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. 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. 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