mathematical physics-icore paper 2: mathematical physics - i unit i vector analysis : gauss...

Mathematical Physics-I

Dr. K.Elampari

Associate Professor of Physics

S.T.Hindu College

Nagercoil

by Dr.K.Elampari, Department of Physics, S.T.Hindu College, Nagercoil-2

It is said that mathematics is the language of Nature.

If so, then physics is its poetry.

• The interplay between mathematics and physics needs no emphasis.

• What may need to be emphasized is that mathematics is not merely atool with which the presentation of physics is facilitated, but the onlymedium in which physics can survive.

• Just as language is the means by which humans can express theirthoughts and without which they lose their unique identity,mathematics is the only language through which physics can expressitself and without which it loses its identity.

• And just as language is perfected due to its constant usage,mathematics develops in the most dramatic way because of its usagein physics

(Hassani Mathematical Physics - A Modern Introduction)by Dr.K.Elampari, Department of Physics, S.T.Hindu

College, Nagercoil-2

Core Paper 2: Mathematical Physics - I

Unit I Vector Analysis : Gauss divergence theorem - Deductions from Gauss divergence

theorem - Green’s theorem - Green’s theorem in a plane - Classification of vector fields.

Unit II Matrices : Eigen values- Eigen Vectors - Characteristic equation of matrix - Cayley

Hamilton theorem - Some important theorems of eigen values and eigen vectors -

diagonalisation of matrices - Differentiation and integration of matrices - Power of matrices

- Exponential of a matrix - Matrices in Physics.

Unit III : Special Functions : Bessel differential equation and Bessels’ function of I Kind -

Generating function - Recurrence relations - Laguarre’s differential equation and Laguerie

polynomial - Generating function - Recurrence relations.

Unit IV : Fourier’s Integral Transforms : Introduction - Fourier’s transform(FT) - Properties of FT - FT of a derivative - Fourier sin and cosine transformsof derivatives - FT of functions of two or three variables - Finite FT - Simpleapplications of FT.

Unit V : Laplace Integral Transforms : Laplace Transform (LT) - Propertiesof LT - LT of derivation of a function - LT of periodic functions - Properties ofinverse LT - Convolution theorem - Evaluation of inverse LT by convolutiontheorem - Application of LT.

Book for Study :

1. Sathya Prakash, Mathematical Physics, Sultan chand & sons, New Delhi.

Books for Reference :

1. Applied mathematics for Engineer and Physics,. Louis A.Pipes LawrenceR.Harvill, Mc Graw Hill Ltd., 1970.

2. Eugune Buthov, Mathematical Physics, Addision Weslay 1968.

Example

• A Plane is flying along , pointing North, but there is a wind coming from the North-West

• There are two Vectors -

Velocity caused by the propeller and

The velocity of the wind

These two vectors interact – the result is the plane slightly slower speed and heading a litter East of North

Subtraction of Vectors

• First is reverse the vector in direction and add them as usual

Components of a Vector

Adding vectors

Subtracting Vectors

Magnitude of a Vector

What is the magnitude of the vector b = (6,8)

|b| = sqrt( 6^2 + 8^2) = 100

Unit Vector

A Vector with Magnitude 1 is called a Unit Vector

Scaling: A Vector can be “scaled” off the unit vector. Here vector ‘a’ is shown to be 2.5 times a unit vector.Notice they still point in the same direction.

Magnitude and Direction in different Perspective.We may know a vector’s magnitude and direction but wants its x and y lengths (or vice versa)

Multiplying a vector by a Scalar

- It is called scaling a vector, because we change how big or how small the vector is.

Multiplying a Vector by a Vector

Dot Product – Directional multiplication, apply the directional growth of one vector on another. The result is how much stronger the original vector ( +ve, -ve or zero)

The projection of a on b (ie) in the direction of b is a Cos ()

Dot product in Cartesiana.b = ax.bx + ay. ByDot product is the interaction between similar dimensions (x * x, y*y, z*z etc)a.b measures, accumulates interaction in matching dimensions.

Calculate a.b

by Dr.K.Elampari, Department of Physics, S.T.HinduCollege, Nagercoil-2

Cross Product a x b

Cross Product accumulates interaction between different dimensions.

• When we need cross product ?

(or) when we need interactions between different dimensions?

• Ans. For example Area

• axb

Assignment

• Properties of Vectors

• Vector Triple Products ( Dot and Cross Product)

Differentiation of vector-valued functions

A curve C is defined by r = r(t), a vector-

valued function of one (scalar) variable. Let us

imagine that C is the path taken by a particle

and t is time. The vector r(t) is the position

vector of the particle at time t and r(t + h) is

the position vector at a later time t + h. The

average velocity of the particle in the time

interval [t, t + h] is then

Partial Derivatives

• If f is any vector function depending upon more than one scalar variable say (x,y,z,t)

• f = f(x,y,z,t)

Scalar and Vector Fields

A function of two or three variables mapping to a scalar is called a

scalar field. one can represent the graph of a scalar field as a curve or

surface.

A function of two or three variables mapping to a vector is called a

vector field.

A vector field F(x, y) (or F(x, y, z)) is often represented by drawing the

vector F(r) at point r for representative points in the domain.

Scalar Field Examples:

Potential fields, such as the Newtonian

gravitational potential, or the electric

potential in electrostatics, are scalar

fields which describe the more familiar

forces. A temperature, humidity or

pressure field, such as those used in

meteorology.

• A good example of a vector field is the

velocity at a point in a fluid;

• at each point we draw an arrow

(vector) representing the velocity (the

speed and direction) of fluid flow

(Figure).

• The length of the arrow represents the

fluid speed at each point.

Gradient of a Scalar Field

The plain old derivative gives us the rate of change of a single variable usually x.

For example dF/dx tells us how much the function F changes in x.

But if a function takes multiple variables x,y, it will have multiple derivatives.Gradient is typically used for functions with several inputs and a singleoutput.

The gradient of a scalar field is the derivative of F in each direction.Therefore, the gradient of a multivariable function has component for eachdirection and the gradient points in the direction of greatest increase.

Therefore, the Gradient of a scalar field is a Vector Field.

• Gradient is a Vector that represents both the magnitude and the direction of maximum space rate of increase of a scalar.

• Watch Gradient of a Scalar Field Video

by Dr.K.Elampari, Department of Physics, S.T.HinduCollege, Nagercoil-2

mathematical physics-icore paper 2: mathematical physics - i unit i vector analysis : gauss...

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