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

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

Geometrical interpretation of Gradient Z P Q dr Y change in f : X =0 => f  dr

Z Q dr P Y X

Magnitude: slope along this maximal direction 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

=> df = 0 for small displacements about the point (x0,y0,z0) If  f = 0 at some point (x0,y0,z0) => df = 0 for small displacements about the point (x0,y0,z0) (x0,y0,z0) is a stationary point of f(x,y,z)

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

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

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

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)

(rate of flow in)EFGH (rate of flow out)ABCD Z X Y dy dx dz A D C B E F H G

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

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

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

Physical significance of Curl Circulation of a fluid around a loop: Y 3 2 4 1 X Circulation (1234)

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

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

Cartesian coordinates in terms of spherical coordinates:

Spherical coordinates in terms of Cartesian coordinates:

Unit vectors in spherical coordinates Z r  Y  X

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

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

Gradient: Divergence:

Curl:

Fundamental theorem for gradient We know df = (f ).dl The total change in f in going from a(x1,y1,z1) to b(x2,y2,z2) along any path: Line integral of gradient of a function is given by the value of the function at the boundaries of the line.

Corollary 1: Corollary 2:

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

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

Electric Dipole E = - V Recall:

Electric Dipole Using:

Fundamental theorem for Divergence Gauss’ theorem, Green’s theorem 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.

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.

THE DIRAC DELTA FUNCTION Recall:

The volume integral of F:

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

From calculation of Divergence: By using the Divergence theorem:

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

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

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

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

The Dirac Delta Function (in one dimension) Properties:

The Dirac Delta Function (in three dimension)

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

So now we can write: Such that: