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## VECTOR CALCULUS

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**multiplication VECTOR**ab b a Vector Product CROSS PRODUCT****• Magnitude : Area of the parallelogram • generatedby a and b. b a Vector Product (contd.)**b**a Vector Product (contd.) • Magnitude :**Direction : Perpendicular to**• both a and b. b a Vector Product (contd.) **b**a Vector Product (contd.) • Direction : A rule is required !!**b**a Right-Hand Rule**b**a • The order of vector multiplication • is important.**A = a b** b sin Area of the parallelogram formed by a and b Geometrical Interpretation**Properties**• Vector multiplication is not • Commutative. • Vector multiplication is Distributive • Multiplication by a scalar**Properties(contd.)**• If and a and b are not null vectors, then a is parallel to b.**j**k i Properties(contd.) Angle between them 0º Angle between them =90º****Vector Product: Components **Examples in Physics**z y x The torque produced by a force is**z**y O x Examples in Physics (contd) The angular momentum of a particle with respect to O**Examples in Physics (contd)**The force acting on a charged particle moving in a magnetic field, Positive charge Negative charge**17**-5 9 • A simple cross product Applications**is perpendicular to**An Example C H E C K**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.**Determine a unit vector perpendicular**to the plane of and Cross product Magnitude Unit vector is An Example**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.**Scalar Triple Product**• Vector Triple Product TRIPLE PRODUCTS****b sin Scalar Triple Product (contd.) Area of the base****a cos = height Scalar Triple Product (contd.) Volume of the parallelopiped**Interchanging any two rows reverses the sign of the**determinant, so • Interchanging rows twice the original sign is restored, so Properties**Properties**• If any two vectors of the scalar triple product are equal, the scalar triple • product is zero.**bac - cab**rule Vector Triple Product**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.**Scalar Product of vectors**a . b = a1 b1 +a2 b2 +a3b3 • Vector Product SUMMARY**Scalar Triple Product : volume of a**• parallelepiped. SUMMARY • Vector Triple Product • Quadruple Product of vectors**Gradient**is a scalar quantity i.e. gradient of a scalar quantity is a vector quantity, Geometrical Interpretation: Gradient has magnitude and direction**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.**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.**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**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.**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**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.