ANALYTICAL TOOLS

VECTORS AND VECTOR ANALYSIS A DIRECTED LINE SEGMENT IS CALLED A VECTOR. THE LENGTH IS CALLED THE LENGTH OF THE VECTOR. AND THE DIRECTION IS CALLED THE DIRECTION OF THE VECTOR. TWO VECTORS ARE EQUAL IF AND ONLY IF THEY HAVE THE SAME LENGTH AND THE SAME DIRECTION. THE LENGTH OF A VECTOR IS CALLED THE EUCLIDEAN NORM OR MAGNITUDE. INNER PRODUCT(DOT PRODUCT) THIS IS ALSO CALLED SCALAR PRODUCT THE RESULT IS A SCALAR a.b = |a||b|cosϒ for a=/= 0, b=/= 0 a.b = 0 a=0 or b=0 and ϒ(0<ϒ<π) IS ANGLE B/W a&b THIS IS COMPUTED WHEN THE VECTORS HAVE THEIR INITIAL POINT COINCIDING. VECTOR PRODUCT ( CROSS PRODUCT) THE RESULT IS A VECTOR Axb (or a^b) v = axb = 0 THIS IS THE CASE IF AND ONLY IF THE VECTORS a AND b HAVE THE SAME OR OPPOSITE DIRECTION OR ONE OF THESE VECTORS IS THE ZERO VECTOR. IN ANY OTHER CASE, v = axb LENGTH = AREA OF PARALLELOGRAM WHOSE SIDES ARE a & b. DIRECTION = PERPENDICULAR TO BOTH a AND b a,b,v FORM A RIGHT HANDED TRIPLE OR RIGHT HANDED TRIAD |v|=|a||b|sinϒ DIRECTIONAL DERIVATIVE THE CONCEPT OF DIRECTIONAL DERIVATIVE HAS TO DO WITH THE DIFFERENTIATION OF A FUNCTION OR A VECTOR FIELD IN A PARTICULAR DIRECTION. THIS GIVES YOU THE RATE OF CHANGE OF THE FUNCTION OR VECTOR FIELD WITH RESPECT TO A GIVEN VARIABLE IN A PARTICULAR DIRECTION. CONSIDER A POINT P IN SPACE AND A DIRECTION AT P GIVEN BY A UNIT VECTOR b LET C BE A RAY AT P IN THE DIRECTION OF b LET Q BE A POINT ON C WHOSE DISTANCE FROM P IS S δf/δs = lim {f(Q) – f(P)}/s s0 s = distance between P and Q IF THE LIMIT EXISTS, IT IS CALLED THE DIRECTIONAL DERIVATIVE OF F AT P IN THE DIRECTION OF b. δf/δs is written as Dbf D = differentiation b= indicates direction From this there are infinitely many directional derivatives of f at P each corresponding to a different direction. Using Cartesian Coordinate system, we may repent the derivative terms of the first partial derivative of f at P in the following manner; If P has a position vector a, The ray C can be represented in the form r(s) = x(s)i + y(s)j + z(s)k (*) = a + sb Where a is the position vector of P s is the distance between P and Q b is the unit vector showing the direction of the ray C therefore δf/δs is the derivative of the function f[x(s),y(s),z(s)] with respect to the arc length s of C Applying the chain rule and assuming that the function f has continuous first partial derivatives δf/δs = (δf/δx)(x’) + (δf/δy)(y’) + (δf/δz)(z’) (‘) denotes derivatives with respect to s and evaluated at s =0 From equation (*) r ’ = X’i + y’j + z’k = b we can introduce the vector grad f = (δf/δx)i + ( δf/δy)j + ( δf/δz)k Using the concept of dot product δf/δs = b.gradf The vector grad f is called the gradient of the scalar f Consider a differential operator V = (δ/δx)i + (δ/δy)j + (δ/δz)k V is called nabla or del grad f = (δf/δx)i + ( δf/δy)j + ( δf/δz)k grad f = (δf/δx)i + ( δf/δy)j + ( δf/δz)k if b has the direction of the positive x axis, then b = i and δf/δs = b. grad f = (δf/δx)i.i = δf/δx Similarly the directional derivative in the positive y-direction is ( δf/δy) etc. Let f(P) = f(x,y,z) be a scalar function having continuous first partial derivatives. Then grad f exists and its length and direction are independent of the particular choice of Cartesian coordinates in space. If at a point P the gradient of “f” is not the zero vector, it has the direction of maximum increase of “f” at P. MAXWELL’S EQUATIONS THESE ARE EQUATIONS WHICH THE FIELDS { E,D,B & H } EVERYWHERE SATISFY.