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**Differential Calculus (revisited):**

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

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**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

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**Geometrical interpretation of Gradient**

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

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Z Q dr P Y X

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**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

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**=> 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)

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

Vector rules Partial differentiation rules

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** 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

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**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.

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**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)

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

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

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

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

DIVERGENCE

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**Curl of a vector is a vector**

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

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**Physical significance of Curl**

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

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

z-component of CURL

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

Spherical coordinates (r, , Ø)

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

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

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**Unit vectors in spherical coordinates**

Z r Y X

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**Line element in spherical coordinates:**

Volume element in spherical coordinates:

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**Area element in spherical coordinates:**

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

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Gradient: Divergence:

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Curl:

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**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.

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Corollary 1: Corollary 2:

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

From the fundamental theorem of gradient: E = - V

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

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Electric Dipole E = - V Recall:

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Electric Dipole Using:

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**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.

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**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.

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**THE DIRAC DELTA FUNCTION**

Recall:

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**The volume integral of F:**

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

From divergence theorem:

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**From calculation of Divergence:**

By using the Divergence theorem:

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

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**The Dirac Delta Function**

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

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**The Dirac Delta Function**

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

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**The Dirac Delta Function**

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

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**The Dirac Delta Function**

(in one dimension) Properties:

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**The Dirac Delta Function**

(in three dimension)

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**The Paradox of Divergence of**

From calculation of Divergence: By using the Divergence theorem:

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So now we can write: Such that:

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Chapter 13-Vector Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

Chapter 13-Vector Calculus Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved.

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