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VECTOR CALCULUS. multiplication VECTOR. a  b. b. a. Vector Product. CROSS PRODUCT. . Magnitude : Area of the parallelogram generated by a and b. . b. a. Vector Product (contd.). b. a. Vector Product (contd.). . Magnitude :. Direction : Perpendicular to

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  2. multiplication VECTOR ab b a Vector Product CROSS PRODUCT

  3. • Magnitude : Area of the parallelogram • generatedby a and b. b a Vector Product (contd.)

  4. b a Vector Product (contd.)  • Magnitude :

  5. Direction : Perpendicular to • both a and b. b a Vector Product (contd.) 

  6. b a Vector Product (contd.)  • Direction : A rule is required !!

  7. b a Right-Hand Rule

  8. b a • The order of vector multiplication • is important.

  9. A = a b  b sin Area of the parallelogram formed by a and b Geometrical Interpretation

  10. Properties • Vector multiplication is not • Commutative. • Vector multiplication is Distributive • Multiplication by a scalar

  11. Properties(contd.) • If and a and b are not null vectors, then a is parallel to b.

  12. j k i Properties(contd.) Angle between them 0º Angle between them =90º

  13. Vector Product: Components 

  14. Vector Product: Components

  15. Vector Product: Components

  16. Vector Product: Determinant

  17. Examples in Physics z y x The torque produced by a force is

  18. z y O x Examples in Physics (contd) The angular momentum of a particle with respect to O

  19. Examples in Physics (contd) The force acting on a charged particle moving in a magnetic field, Positive charge Negative charge

  20. 17 -5 9 • A simple cross product Applications

  21. is perpendicular to An Example C H E C K

  22. Find a unit vector perpendicular to the plane containing two vectors • A vector perpendicular to a • and b is • Corresponding unit vector Applications(contd.) • finding a unit vector • perpendicular to a plane.

  23. Determine a unit vector perpendicular to the plane of and Cross product Magnitude Unit vector is An Example

  24. SUMMARY a. b = ax bx +ay by +azbz • Scalar product • Magnitude and Direction of a vector remain invariant under transformation of coordinates. • Product of a vector with a scalar is a vector quantity • Vector product : directional property, • denotes an area.

  25. Scalar Triple Product • Vector Triple Product TRIPLE PRODUCTS

  26. Scalar Triple Product

  27. Scalar Triple Product (contd.)

  28. Scalar Triple Product (contd.)

  29. b sin Scalar Triple Product (contd.) Area of the base

  30. a cos  = height Scalar Triple Product (contd.) Volume of the parallelopiped

  31. Interchanging any two rows reverses the sign of the determinant, so • Interchanging rows twice the original sign is restored, so Properties

  32. Properties • If any two vectors of the scalar triple product are equal, the scalar triple • product is zero.

  33. bac - cab rule Vector Triple Product

  34. SUMMARY • A physical quantity which has • both a magnitude and a direction • is represented by a vector • A geometrical representation • An analytical description: components • Can be resolved into components along any three directions which are non planar.

  35. Scalar Product of vectors a . b = a1 b1 +a2 b2 +a3b3 • Vector Product SUMMARY

  36. Scalar Triple Product : volume of a • parallelepiped. SUMMARY • Vector Triple Product • Quadruple Product of vectors

  37. Gradient is a scalar quantity i.e. gradient of a scalar quantity is a vector quantity, Geometrical Interpretation: Gradient has magnitude and direction

  38. For fix value of magnitude of dl, df is greatest when cos  is zero, i.e. we move in the same direction as f. • The gradient f points in the direction of maximum increase of the function f. • The magnitude f gives the slope (rate of increase) along this maximal direction.

  39. Divergence is a vector quantity. is a scalar quantity is known as divergence of a vector quantity ( ) Physical Significance It represents how much the vector spreads out ( diverges) from the point. If divergence of any vector is positive then it shows Spreading out and if negative then coming towards that point.

  40. Point works as Source Point works as Sink Eg: Divergence of current density: (Current density : current per unit area) at a point gives the amount of charge flowing out per second per unit volumefrom a small closed surface surrounding the point. i.e. the flux entering any element of space is exactly balanced by that leaving it. Such vectors are known as solenoidal vector

  41. Curl is a vector quantity is a vector quantity known as curl of • Physical Significance It is a measure of how much the vector curls around the point.

  42. Problems • 1. If A=3x2y - y3x2, calculate gradient A at a point (1,-2,-1) • 2. If = x2yi-2xzj+2yzk, calculate divergence and curl of a vector at (1,2,1). Ans: 1. 10i-9j 2.(i) 6 (ii) k

  43. Second Derivatives • The gradient, the divergence and the curl are the only first derivatives we can make with , by applying twice we can construct five species of second derivatives. • The gradient is a vector, so we can take the divergence and curl of it. (1) Divergence of gradient : (Laplacian) (2) Curl of gradient: • The divergence is a scalar, so we can take its gradient. (3) Gradient of divergence. • The curl is a vector, so we can take its divergence and curl. (4) Divergence of a Curl. (5) Curl of curl.

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