2. jeo.;a kshï; fj,djg foaYk / ksnkaOk mx;s i|yd meïfkkak. fyd|ska ijka fokak. úIh wdYs%; fmd;a lshjkak. úIh wdYs%;.egˆ úi|kak.

4. 1. Scalar Fields. 2. Directional Derivative. 3. Systems of Coordinates. 4. Curvilinear Coordinates. 5. Integral along a curve. 6. Green's Theorem in a plane. 7. Surface Integrals and Surface area. 8. Volume integrals. 9. Stokes Theorem. 10. The Divergence Theorem. 11. Applications of Vector Analysis in Classical Field Theory. Course Content:

5. Method of Assessment : End of Semester One hour written examination 70% Mid Term Examination 10% Computer Practicals 20% Tutorials will be given. Reference  Calculus by James Stewart  Calculus and Analytic Geometry (Addison & Wesley) Thomas, G. B. & Finney, R. L.  Vector Calculus by R Guptha.  Any other Calculus book.

6. w¾: oelaùu - ugsgus mDIaG  =  ( x, y, z, t ) wosY fCIa;%h i|yd  = c ksh;hla u.ska,efnk mDIaGj,g ugsgus mDIaG hehs lshkq,efns. Definition – Level surfaces F or the scalar field  =  ( x, y, z, t ), the surfaces given by  = c, are called level surfaces. MAT 128 1.0 Mathematical Tools and Computer Practicals II 1. wosY fCIa;% / Scalar Fields. w¾: oelaùu wjldYfha huÞ m%foaYhl we;s iEu,laIHhla yd ix>Ü;j wosYhla w¾: olajd we;s úg thg wosY lafIa;%hla hehs lshkq,efnÞ. Definition Suppose a scalar is defined associated with each point in some region of space, then it is called a scalar field.

7. Consider the scalar field  = x 2 + y 2 + z 2 + t 2. When t = t o, the level surfaces are given by x 2 + y 2 + z 2 +t 2 o = c. i.e. x 2 + y 2 + z 2 = k. When k = c - t 0  0, it represents a system of spheres. k = c - t 0  0 ú g fuhska f.da, moaO;shla ksrEmKh fjs.

8. Two dimensional situation When the scalar field is defined on a plane, by considering a system of Oxy axis in the plane, we can represent the scalar field as  =  (x, y, t). Then the equation  (x,y,t 0 ) = c represents the level curves. Eg.  = 2t( x 2 -4xy 2 ) Then the level curves are 2 t 0 ( x 2 - 4 x y 2 ) = c i.e x 2 – 4 x y 2 = k

9. Consider the points P=(x, y, z) in the region where scalar field  is defined. Let Q= (x+  x, y+  y, z+  z) be an arbitrary closed point in the same region. Then of change of the scalar field when we move from P to Q in the straight path is Directional Derivative (osYd jHq;amkakh&, and called Difference Quotient. So rate of change of the scalar field at P along the direction PQ is This limiting value is defined as the Directional derivative of  at P in the direction of PQ.

10. Notation : If is the unit vector in the direction of PQ, Y X O Z    Q P xx yy zz Note : Directional derivative is a function of time. we denote the directional derivative of  by

11. Here PQ 2 = (x +  x –x) 2 + (y +  y –y) 2 + (z +  z –z) 2 =  x 2 +  y 2 +  z 2    Q P xx yy zz

13. So directional derivative of the scalar field at P in the direction PQ is E.g. For the scalar field find the directional derivative when t = t o i ) at ( 1, 2, 1) towards the point ( 3, 2, 0). ii) at ( 3, 1, 2) in the direction

14. Solution i )

15. ii).

16. E.g. For the scalar field find the directional derivative i ) at ( 1, 2, 0) towards the point ( 3, 2, 1). ii) at ( 0, 1, 2) in the direction Solution i) at ( 1, 2, 0), directional derivative is ii) at ( 0, 1, 2), directional derivative is

17. Note: If the directional derivative is positive then the scalar field increases as going from P to Q along the given direction. If the directional derivative is negative then the scalar field decreases as going from P to Q along the given direction. Properties of the Directional Derivative

18. P So in the perpendicular direction to ;the value of the scalar field does not change. So we get the maximum value of the directional derivative along the direction of the gradient and minimum in the opposite direction.

19. E.g. Find the direction such that directional derivative of the scalar field does not change. is maximum. at ( 3, 1, 0 ) Solution Scalar field would not change in any direction perpendicular to Letbe such a direction. Then

20. and b is arbitrary. Directional derivative is maximum along the direction

21. For the scalar field , the directional derivative at P in the direction of l is gradient is. P ysoS l osYdjg  wosY fCIa;%fha osYd jHq;amkakh fjs. Summery