INTRODUCTION TO VECTOR CALCULUS

Transcription

1 INTRODUTION TO VETOR LULUS LSSIFITION OF VETORS roadly vectors can be classified into two categories (i) (ii) xial Vectors : Where a vector has rotational motion lying along the normal to the plane of rotation of the body and remains unchanged under inversion. e.g.: Torque, angular momentum etc. Polar Vector : Where a vector has linear motion in a particular direction but changes under inversion or reflection. e.g. displacement, position vector, velocity etc. Some special vectors (i) Unit vector : It is a vector with unit magnitude and characterizes the direction of the vector mathematically it is denoted by  = In artesian coordinate system, let us choose three unit vectors along three mutually perpendicular axes as ˆi, ˆj and ˆk in x, y and z directions respectively. Then any arbitrary vector can be expressed as = x i ˆ ˆ ˆ y j zk where x, y and z are called the components of in x, y and z directions.

2 (2) Introduction to Vector alculus y y ˆk ĵ î x ˆ zk 0 ˆ y j x z z xˆ i The magnitude of is given using parallelogram law = hence unit vector along x y z is given by xiˆ ˆ y j ˆ zk  = Direction cosines The cosines of the angles, which makes with x, y and z axis are called direction cosines of the vector. If l, m and n are the direction cosines along ox, oy and oz axes, then l = x x cos or l m = n = y cos or y m z cos or z n and l m n = 2 x y z 2 2 1

3 Introduction to Vector alculus (3) so l m n = cos cos cos 1 li ˆ m ˆ j n k ˆ and = and the unit vector â = liˆ mˆj n kˆ (ii) (iii) Null Vector : ny vector with magnitude zero is called null vector. It is collinear with every vector and denoted by O. ollinear or parallel vector : When vectors are parallel, then these are collinear vectors, whatsoever their magnitudes may be. Direction of there vectors may be some or opposite. (iv) (v) (vi) D When any scalar is multiplied to any vector then the resultant vector becomes collinear with original one. e.g. i.e. vector is times with same direction as of. oplanar vectors : When vectors lies in the same geometrical plane they are called coplanar vectors. Otherwise these are called non-coplanar vectors. Like vectors : The collinear vectors with same sense of direction irrespective of magnitude are called like vectors. Reciprocal vectors : When the magnitude of a vector is reciprocal to the magnitude of other vector with same direction then it is called reciprocal vector. It is written as 1 i.e. 1 â where â is the unit vector along the direction of. Product of vectors (i) Scalar product or dot product : When the result of product of two vectors is a scalar quantity then this product is known as scalar (or dot) product of the given vectors.

4 (4) Introduction to Vector alculus Mathematically it is obtained by multiplying the magnitudes of the vectors with cosines of the angle between them i.e.. = cos. = cos P 0 y cos Q lternatively scalar product may be defined as multiplication of one vector with component of another in the direction of first. In case of artesian unit vectors ˆi.i ˆ = i. i cos ˆj. ˆj k ˆ. kˆ and ˆi. ˆj = i j cos 90 0 ˆj. kˆ i ˆ. kˆ if = x i ˆ ˆ ˆ y j z k and = ˆ x i ˆ ˆ y j z k then. x x y y z z (a) Scalar product obeys commutative law i.e... (b) It obeys distributive law i.e.... (c) Two non-zero vectors are orthogonal or perpendicular when = 90 i.e. cos = 0 then. = 0 similarly two vectors are collinear when = 0 or i.e. cos = ± 1 then for = 0,. = and for =,. = Physical examples (i) Work done W F. ds

5 Introduction to Vector alculus (5) (ii) (ii) Power = F.v (iii) Magnetic flux of a magnetic field =. ds where is magnetic flux density over an area ds. (iv) Electric flux of an electric field = E. ds where E the electric field intensity through elementary area ds. Vector product or cross product When the product of two vectors is a vector quantity, then the product is called vector product or cross product mathematically it is written as = = sin. nˆ where 0 here ˆn is the unit vector in the direction of normal to the plane containing and such that, and from a right handed coordinate system with rotation from to. for ˆi, ˆj and k ˆ. ˆi ˆi = i i sin 0 0 ˆj ˆj kˆ kˆ and ˆi ˆj = i j sin90 1 ˆj kˆ kˆ ˆi but ˆj ˆi = j i sin 90 1 kˆ ˆj ˆi kˆ if = ˆ xi ˆ ˆ y j z k and = xi ˆ ˆ ˆ y j z k then = sin n ˆ or ˆ ˆ ˆ i j k = = yz zy iˆ ˆ xz zx j xy yx

6 (6) Introduction to Vector alculus and ˆn = hence sin = where = and = x y z x y z if the rotation from to is anti clockwise then is +ve. and if rotation is clockwise then is ve. 0 Properties (a) ross product is not commutative but (b) It is distributive i.e. (c) If two vectors are collinear or parallel then = 0 or then sin = 0 and = 0 (d) Two vectors are perpendicular then = 90 so = nˆ

7 Introduction to Vector alculus (7) Some examples (i) Moment of forces = r F L r p m r v (iii) Linear velocity v r (ii) ngular momentum (iv) (v) Scalar Triple product force on a charged particle F = qv where q is in coulombs. Force on a charged particle moving through electric and magnetic field is q E v. this is known as Lorentz force. F = When a vector is scalarly multiplied with the cross product of other two vectors then the result is called scalar triple product. =.... = Vector triple product When any vector is vectorily multiplied with vector product of other two vectors taken in cyclic order then the result is known as vector triple product... Properties = =.. =.. 0 (i) (ii)..

8 (8) Introduction to Vector alculus (iii). D..D.D. D. D. D (iv) Vector differentiation This is the limiting value of ratio of a vector to the change of a scalar as the change tends to zero is called vector differentiation. f (u + u) f = f (u + u) f (u) Properties (a) f (u) df f = lim du u 0 u u u f u = lim u 0 u d d d du du du d d... d du du du d d d du du du d d d d.... du du du du d d d d du du du du d da ds as dt ds dt (b) (c) (d) (e) (f)

9 Introduction to Vector alculus (9) d de e.e e. de e 2e de dt dt dt dt d 2 de e 2e. dt dt d da at dt dt (g) (h) or