“Wishing you a very happy and successful new year-2015”

Welcome to PYL100 course

Lecture-2 on 06/01/2015 By: Rajendra S. Dhaka (rsdhaka@physics.iitd.ac.in)  

PYL100: Electromagnetic Waves and

Quantum Mechanics 1  

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Ch.1: The Divergence: Brief overview

Ø From the definition of , we can construct the dot product of with a vector, called

divergence

Ø  the divergence of a vector function is itself a scalar..

Ø  we cannot have the divergence of a scalar Ø  it tells us how much the function v spreads

out (diverges) from the point in question.

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The Divergence

Ø  The mathematical definition of divergence is:

where the surface S is a

closed surface that completely surrounds a very small volume Δv at point r , and where ds points outward from the closed surface.

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The Divergence

² From the definition of surface integral, divergence basically indicates the amount of vector field A(r) that is converging to/diverging from, a given point.

² For example, consider these vector fields in the region of a specific point:

Δv  Δv  

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Ø  The field on the left is converging to a point, Ø  Therefore, the divergence of the vector field at that

point is negative. Ø Conversely, the vector field on the right is diverging

from a point. Ø As a result, the divergence of the vector field at that

point is greater than zero. Ø Consider some other vector fields in the region of a

specific point:

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The Divergence:

² Geometrical Interpretation: a measure of how much

the vector field spread out at the point where the derivatives are evaluated: ² Examples:

Ø  Taps/fountain: points of +ve divergence Ø  Sink/drain: points of –ve divergence

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The Curl:

² From the definition of , we can construct— the cross product of with a vector, called the curl: which, when we expend, yields the following---

Ø Curl creates another vector out of the vector field

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The Curl:

It means:

a measure of how much the vector “curls around” the point in question or how much is the rotational effect.

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The Curl:

² A magnetic field has the property

² An electrostatic field has the property

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Integral Calculus: Line integral

is vector function

is infinitesimal displacement vector

v Integral is to be carried out along a path P from a-b.      

² At each point on the path, we take the dot product of with the displacement to the next point on the path.

Ø  For a closed loop path:

² An example: work done by a force

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Integral Calculus: Surface integral is vector function

is infinitesimal path area, with direction perpendicular to the surface

² For closed surface:

Ø  for closed surface, outward is positive

Ø  for open surface, it is arbitrary.

If represent the flow of liquid (i.e. mass per unit area per unit time)

Then = total mass per unit time passing through the surface = flux

S

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Integral Calculus: volume integral T is scalar function

dτ is infinitesimal volume element

² In Cartesian coordinates: dτ = dxdydz

Ø  For example, if T = T(x, y, z) = density of substance, then…….

The fundamental theorem of calculus:

² Suppose f(x) is a function of one variable: the fundamental theorem of calculus states:

v

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The fundamental theorem of calculus:

² Note that the value of this integral depends only on the value of the functions f(x) at the end points of the integral and does not depend on how the function varies in between

² If you chop the interval from a to b into many tiny pieces, dx, and add up the increments df from each little piece, the result is equal to the total change in f

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The fundamental theorem for Gradients:

² Suppose we have a scalar function of three variables T(x, y, z)

starting at point a, we move a small distance dl1, the function T will change by an amount

total change in T

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The fundamental theorem for Divergences: [Divergence theorem or Gauss’s (Green’s) theorem]

² It states that:

² If v = flow of incompressible fluid

= total amount of fluid passing out through the surface per unit time.

Ø Divergence measures the “spreading out” of the vectors from a point—

Ø  a place of high divergence is like a “faucet” pouring out liquid…& if there are many faucets..

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The fundamental theorem of Curls: (Stoke’s theorem)

The integral of a curl over some surface or the flux of the curl through that surface represents the “total amount of swirl”

Flux of curl through the surface Line integral of vector around = Total amount of rotation the boundary

This can be determined by going around the edge & finding how much the flow is following the boundary

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Second Derivatives:

² The gradient, the divergence, and the curl are first derivatives, which use

² If we apply twice, we can construct five species of second derivatives.

² The gradient is a vector, so we can take the divergence and curl of it:

² The divergence is a scalar—all we can do is to take gradient---

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Second Derivatives:

² The curl is a vector, so we can take its divergence and curl:

² Divergence of gradient of a vector is:

unlike del the operator, is a scalar, it is called the Laplacian.

Ø  It can operate on both scalars and vectors.

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Second Derivatives:

² Curl of the gradient: this follows from our definition of curl….

² Gradient of divergence:

² Please also note the following:

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Other self-study topics:

² Cartesian Coordinates:

² 1.4 Curvilinear Coordinates:

Ø  1.4.1 Spherical Polar Coordinates

Ø  1.4.2 Cylindrical Coordinates

² Also, please read the following:

Gradient, Divergence and Curl in the above mentioned different coordinate systems…… *************END*****************