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Derivative of any function f(x,y,z): Differential Calculus (revisited): Gradient of function f

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Gradient of a function Change in a scalar function f corresponding to a change in position dr f is a VECTOR

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Geometrical interpretation of Gradient Z X Y PQ dr change in f : =0 => f dr

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Z X Y P Q dr

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For a given |dr|, the change in scalar function f(x,y,z) is maximum when: => f is a vector along the direction of maximum rate of change of the function Magnitude: slope along this maximal direction

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If f = 0 at some point (x 0,y 0,z 0 ) (x 0,y 0,z 0 ) is a stationary point of f(x,y,z) => df = 0 for small displacements about the point (x 0,y 0,z 0 )

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The Operator is NOT a vector, VECTOR OPERATOR but a VECTOR OPERATOR Satisfies:Vector rules Partial differentiation rules

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On a scalar function f : f can act: GRADIENT On a vector function F as:. F DIVERGENCE On a vector function F as: × F CURL

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Divergence of a vector.F is a measure of how much the vector F spreads out (diverges) from the point in question. Divergence of a vector is a scalar.

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Physical interpretation of Divergence Flow of a compressible fluid: (x,y,z) -> density of the fluid at a point (x,y,z) v(x,y,z) -> velocity of the fluid at (x,y,z)

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Z X Y dy dx dz A DC B E F HG (rate of flow in) EFGH (rate of flow out) ABCD

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Net rate of flow out (along- x) Net rate of flow out through all pairs of surfaces (per unit time):

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Net rate of flow of the fluid per unit volume per unit time: DIVERGENCE

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Curl Curl of a vector is a vector ×F is a measure of how much the vector F curls around the point in question.

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Physical significance of Curl Circulation of a fluid around a loop: X Y Circulation (1234)

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Circulation per unit area = ( × V )| z z-component of CURL

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Curvilinear coordinates: used to describe systems with symmetry. Spherical coordinates (r,, Ø )

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Cartesian coordinates in terms of spherical coordinates:

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Spherical coordinates in terms of Cartesian coordinates:

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Unit vectors in spherical coordinates r Z X Y

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Line element in spherical coordinates: Volume element in spherical coordinates:

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Area element in spherical coordinates: on a surface of a sphere (r const.) on a surface lying in xy-plane ( const.)

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Gradient: Divergence:

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Curl:

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Fundamental theorem for gradient We know df = ( f ). dl The total change in f in going from a(x 1,y 1,z 1 ) to b(x 2,y 2,z 2 ) along any path: Line integral of gradient of a function is given by the value of the function at the boundaries of the line.

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Corollary 1: Corollary 2:

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Field from Potential From the definition of potential: From the fundamental theorem of gradient: E = - V

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Electric Dipole Potential at a point due to dipole: z y x p r

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Electric Dipole E = - V Recall:

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Electric Dipole Using:

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Fundamental theorem for Divergence The integral of divergence of a vector over a volume is equal to the value of the function over the closed surface that bounds the volume. Gauss theorem, Greens theorem

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Fundamental theorem for Curl Stokes theorem Integral of a curl of a vector over a surface is equal to the value of the function over the closed boundary that encloses the surface.

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THE DIRAC DELTA FUNCTION Recall:

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The volume integral of F:

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Surface integral of F over a sphere of radius R: From divergence theorem:

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From calculation of Divergence: By using the Divergence theorem:

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Note: as r 0; F And integral of F over any volume containing the point r = 0

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The Dirac Delta Function (in one dimension) Can be pictured as an infinitely high, infinitesimally narrow spike with area 1

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The Dirac Delta Function (x) NOT a Function But a Generalized Function OR distribution Properties:

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The Dirac Delta Function (in one dimension) Shifting the spike from 0 to a;

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The Dirac Delta Function (in one dimension) Properties:

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The Dirac Delta Function (in three dimension)

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From calculation of Divergence: By using the Divergence theorem: The Paradox of Divergence of

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So now we can write: Such that:

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