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

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

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Differential Calculus (revisited):

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

Gradient of function f


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


For a given |dr|, the change in scalar function f(x,y,z) is maximum when:

=> f is a vectoralong the direction of maximum rate of change of the function

Magnitude: slope along this maximal direction


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

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)


Z

G

H

C

D

dz

E

F

dx

Y

A

B

dy

X

(rate of flow in)EFGH

(rate of flow out)ABCD


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

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:


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:


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