Mathematics is the language of physics. It permits us to discuss the laws of physics and their consequences in an attractively simple and compact manner. As Galileo so delightfully expressed: "The great book of nature lies ever open before our eyes, but it is written in mathematical characters." Experience tells us that there has been a fascinating interplay between physics and mathematics all along.
To understand and express the concepts and laws of physics, it is essential that you learn the relevant mathematical techniques. For this purpose, IGNOU are offering two 2-credit courses PHE-04 and PHE-05 on Mathematical Method in Physics. In the PHE-04 course you will study Vector Calculus (Block 1), and Probability and Statistics (Block 2).
The study of vector calculus is of great value in physics. While the use of vectors simplifies the expression of physical concepts, their differentiation and integration is useful in determining the resultant of forces or momenta, determining the work done on a particle moving in any force field, modeling the flow of fluids in pipes, computing the electric field due to a charged conductor, etc. Indeed, the tools of vector calculus discussed in Block 1 find extensive use in mechanics, electromagnetics, quantum mechanics, fluid mechanics, optics, etc.
There is perhaps no part of mathematics that is more intimately connected with everyday experiences than probability theory. We all know that the element of chance dominates the physical world. Even the origin of life is itself a chance happening! Probability theory serves as a model for chance phenomena. Along with statistical methods, it enables us to deal with any problem involving large number of particles/variables. Knowledge of probability and statistics is required in such diverse areas as the kinetic theory of matter, statistical mechanics, quantum mechanics, atmospheric physics. design of experiments, interpretation of data and error analysis.
If you intend to study physics in depth, you must take this course, particularly because vector calculus is not being discussed in any of the mathematics courses.
Syllabus Per Block Divisions in IGNOU BSc Physics - Mathematical Methods in Physics-I PHE-04
Block 1: Vector Calculus
- Unit I: Vector Algebra - Vectors and scalars, Vector Components relative to a Coordinate System, Transformation of Coordinate Systems and Vector Components, Precise Definition of Three- dimensional Vectors; Analytical Approach to Vector Algebra: Vector Addition and Subtraction, Scalar and Vector Products in Component Form; Multiple Product of Vectors, Scalar and Vector Triple Products, Quadruple Products of Vectors; Polar and Axial Vectors.
- Unit 3: Coordinate Systems - Non-Cartesian Coordinate Systems: Plane Polar Coordinates, Spherical and Cylindrical Polar Coordinates; Expressing a Vector, Differential Elements of Vector Length and Area and Gradient in Polar Coordinates; Generalised Curvilinear Coordinate System: Orthogonality Condition, Unit Vectors, Vector Differential Operators, i.e., Gradient, Divergence, Curl and in Different Coordinate Systems.
- Unit 4: Integration of Scalar and Vector Fields - Vector Integration: Integration of a Vector with respect to a Scalar: Integrals involving Scalar and Vector Products of Vectors; Multiple Integrals: Double and Triple Integrals; Line Integral of a Field: Path of Integration, Types of Line Integrals, Evaluation of Line Integrals; Surface Integral of a Field: Surface of Integration, Types of Surface Integrals; Evaluation of Surface Integrals, Volume Integral of a Field: Types of Volume Integrals, Evaluation of Volume Integral; Vector Integral Theorems: Gauss' Divergence Theorem, Stoke's Theorem, Green's Theorem (Statement only and Physical Meaning), Applications in Physics.
- Appendix: Proof of the Vector Integral Theorems
Block 2: Probability and Statistics
- Unit 5: Basic Concepts of Probability
- Unit 6: Probability Distributions
- Unit 7: Applications in Physics
Elements of Probability Theory: Basic Terminology, Elementary Combinatorics, Fundamental Theorems (Theorems of Probability, Conditional Probability and Total Probability Theorem, Bayes' Theorem); Random Variable; Expectation and Variance, Covariance and Correlation Coefficient, Binomial and Poisson Distributions, Normal Distribution, Continuous Probability Distributions in Physics; Correlation Analysis; Regression Analysis: Method of Least Squares, Standard Error of Estimate; Statistical Theory of Errors: Types of Errors, Estimation of Random Errors for a Single Variable, Random Error Estimation for Indirect Measurements.