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

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

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Change in a scalar function f corresponding to a change in position dr

f is a VECTOR

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)

Net rate of flow out (along- x)

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

Curl

Curl of a vector is a vector

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

Circulation per unit area = ( × V )|z

z-component of CURL

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.)

Divergence:

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 2:

Field from Potential

From the definition of potential:

From the fundamental theorem of gradient:

E = - V

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.

Recall:

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

(in one dimension)

Can be pictured as an infinitely high, infinitesimally narrow “spike” with area 1

(x) NOT a Function

But a Generalized Function OR distribution

Properties:

(in three dimension)

Such that:

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