Download presentation

Presentation is loading. Please wait.

1
**Differential Calculus (revisited):**

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

2
**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

3
**Geometrical interpretation of Gradient**

Z P Q dr Y change in f : X =0 => f dr

4
Z Q dr P Y X

5
**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

6
**=> 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)

7
**The Operator is NOT a vector, but a VECTOR OPERATOR Satisfies:**

Vector rules Partial differentiation rules

8
** 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

9
**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.

10
**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)

11
(rate of flow in)EFGH (rate of flow out)ABCD Z X Y dy dx dz A D C B E F H G

12
**Net rate of flow out (along- x)**

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

13
**Net rate of flow of the fluid per unit volume per unit time:**

DIVERGENCE

14
**Curl of a vector is a vector**

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

15
**Physical significance of Curl**

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

16
**Circulation per unit area = ( × V )|z**

z-component of CURL

17
**Curvilinear coordinates: used to describe systems with symmetry.**

Spherical coordinates (r, , Ø)

19
**Cartesian coordinates in terms of spherical coordinates:**

20
**Spherical coordinates in terms of Cartesian coordinates:**

21
**Unit vectors in spherical coordinates**

Z r Y X

22
**Line element in spherical coordinates:**

Volume element in spherical coordinates:

23
**Area element in spherical coordinates:**

on a surface of a sphere (r const.) on a surface lying in xy-plane ( const.)

24
Gradient: Divergence:

25
Curl:

26
**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.

27
Corollary 1: Corollary 2:

28
**E = - V Field from Potential From the definition of potential:**

From the fundamental theorem of gradient: E = - V

29
Electric Dipole Potential at a point due to dipole: z r p y x

30
Electric Dipole E = - V Recall:

31
Electric Dipole Using:

32
**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.

33
**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.

34
**THE DIRAC DELTA FUNCTION**

Recall:

35
**The volume integral of F:**

36
**Surface integral of F over a sphere of radius R:**

From divergence theorem:

37
**From calculation of Divergence:**

By using the Divergence theorem:

38
Note: as r 0; F ∞ And integral of F over any volume containing the point r = 0

39
**The Dirac Delta Function**

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

40
**The Dirac Delta Function**

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

41
**The Dirac Delta Function**

(in one dimension) Shifting the spike from 0 to a;

42
**The Dirac Delta Function**

(in one dimension) Properties:

43
**The Dirac Delta Function**

(in three dimension)

44
**The Paradox of Divergence of**

From calculation of Divergence: By using the Divergence theorem:

45
So now we can write: Such that:

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google