We now begin with vector calculus which concerns two kinds of functions: Vector functions: Functions whose values are vectors depending on the points

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1 94 Vector and scalar functions and fields Derivatives We now begin with vector calculus which concerns two kinds of functions: Vector functions: Functions whose values are vectors depending on the points v v( ) v ( ) iv ( ) jv ( ) k in space, Scalar functions: Functions whose values are scalars depending on the points in space, f f( ) Here, is a point in the domain of definition, which in applications is a 3-D domain or a surface or a curve in space A vector function defines a vector field and a scalar function defines a scalar field in that domain or on that surface or curve Examples of vector fields are field of tangent vectors of a curve, field of normal vectors of a surface, velocity field of a rotating body and the gravitational field (see Figs ) Examples of scalar fields are the temperature field in a body or the pressure field of the air in the earth's atmosphere Vector and scalar fields may also depend on time t or on some other parameters If we introduce Cartesian coordinates x, yz,, then instead of v( ) we can write, v v( x, yz, ) v( xyz,, ) iv( xyz,, ) jv( xyz,, ) k Note that the components depend on the choice of the coordinate system, whereas a vector field that has a physical or a geometric meaning should have magnitude and direction depending only on, not on the choice of coordinate system Similarly for the value of a scalar field f ( ) f( x, y, z) Example 1: Scalar function (Euclidean distance in space) The distance f ( ) of any point from a fixed point in space is a scalar function whose domain of definition is the whole space f ( ) defines a scalar field in space If a Cartesian coordinate system is introduced and has the coordinates x, y, z and has the coordinates x, yz, then f is given by, f( ) f( x, y, z) [( xx ) ( y y ) ( zz ) ] 1/ If the given Cartesian coordinate system is replaced by another Cartesian coordinate system obtained by translating and rotating the given system, then the values of the coordinates of and will in general change, but f ( ) will have the same value as before Hence f ( ) is a scalar function June 5,

2 Example : Vector field (Velocity field) At any instant the velocity vectors v( ) of a rotating body B constitutes a vector field, called the velocity field of the rotation If a Cartesian coordinate system having the origin on the axis of rotation is introduced then (see Ex 5, Sec 93), v( x, yz, ) wr w( xi yj zk ), (941) where x, yz, are the coordinates of any point of B at the instant under consideration If the coordinate system is such that the z -axis is the axis of rotation and w points in the positive z -direction, then w k and i j k v ( yixj ) x y z An example of a rotating body and the corresponding velocity field are shown in Fig 195 Example 3: Vector field (Field of force, gravitational field) A particle A of mass M is fixed to a point and a particle B of mass m is free to take up various positions in space Then particle A attracts particle B According to Newton's law of gravitation the corresponding gravitational force p is directed from to, and its magnitude is proportional to 1/r, where r is the distance between and, c p, (94) r 8 3 where c GMm and G 6671 cm /(gm sec ) is the gravitational constant Hence p defines a vector field in space If a Cartesian coordinate system is introduced such that has the coordinates x, y, z and has the coordinates x, yz, then, r [( xx ) ( y y ) ( zz ) ] 1/ Assuming that r and introducing the vector, r ( x x ) i( y y ) j( zz ) k, June 5,

3 it is seen that r r and ( 1/ r) r is a unit vector in the direction of p ; the minus sign indicated that p is directed from to From this and eqn (94), 1 c c p p ( ) r r [( xx 3 3 ) i( yy) j( zz ) k ] (943) r r r This vector function describes the gravitational force acting on B 941 Vector calculus Next it is shown that the basic concepts of calculus, ie, convergence, continuity, and differentiability can be defined for vector functions in a simple and natural way Most important here is the derivative Convergence: An infinite sequence of vectors a ( n ), n 1,,, is said to converge if there is a vector a such that lim a a (944) n ( n) Here a is called the limit vector of that sequence, and we write, lima a (945) n ( n ) Cartesian coordinates being given, this sequence of vectors converge to a if and only if the three sequences of components of the vectors converge to the corresponding components of a Similarly, a vector function v( t) of a real variable t is said to have the limit l as t approaches t, if ( t) is defined in some neighborhood of (possibly except at t ) and v t lim v( t) l (946) tt Then we write, lim v( t) l (947) tt Here a neighborhood of t is an interval (segment) on the t -axis containing t as an interior point (not as an endpoint) June 5,

4 Continuity: A vector function v() t is said to be continuous at t t if it is defined in some neighborhood of (including at t itself) and t lim v( t) v ( t ) (948) tt If a Cartesian coordinate system is introduced, we may write, v() t v () t iv () t jv () t k Then () t is continuous at if and only if its three components are continuous at t v t Derivative of a vector function: A vector function the following limit exists, v() t is said to be differentiable at a point t if v() t lim v( tt) v( t t This vector v ( t) is called the derivative of v( t) See fig 197 t ) (949) In components with respect to a given Cartesian coordinate system, v() t v() t iv() t jv() t k (941) Hence the derivative v ( t) is obtained by differentiating each component separately For instance, if v() t tit j then v( t) itj Equation (941) follows from (949) and conversely because (949) is a "vector form" of the usual formula for calculus by which the derivative of a function of a single variable is defined (The curve in Fig 197 is the locus of the terminal points representing v( t) for values of the independent variable in some interval containing t and t t in (949)) The familiar differentiation rules continue to hold for differentaiating vector functions, for instance, and in particular, ( cv) cv, uv, u vuv, (9411) uvuv, (941) u( vw) u( vw) u( vw) u( vw ) (9413) June 5,

5 Example 4: Derivative of a vector function of constant length Let v( t) be a vector function whose length is constant, say, v() t c Then v v v c, and ( vv) vv, by differentiation Hence, the derivative of a vector function v() t of constant length is either the zero vector or is perpendicular to v() t 94 artial derivatives of a vector function artial differentiation of vector functions of two or more variables can be introduced as follows Suppose that the components of a vector function, v() t v () t iv () t jv () t k, are differentiable functions of n variables t,, 1 tn Then the partial derivative of v with respect to t is denoted by v m / t m and is defined as the vector function, v v1 v v3 i j k t t t t m m m m Similarly, second partial derivatives are and so on v v1 v v3 i j k, t t t t t t t t l m l m l m l m Example 5: artial derivatives r Let r( t1, t) acost1iasint1j tk Then asin t1i acost1j and t 1 r t k Various physical and geometric applications of derivatives of vector functions will be discussed in the next sections and in Chapter 1 June 5,

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