Download presentation

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

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google