CSIR NET Physical Sciences Mathematical Methods of Physics Syllabus

Mathematical Methods of Physics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematical Methods of Physics in CSIR NET Physical Sciences, not a summary of it.

1. 1

   Dimensional Analysis

   overview

   Examined as a single unit within Mathematical Methods of Physics — no further topic split in the official outline.

2. 2

   Vector Algebra and Vector Calculus

   overview

   Examined as a single unit within Mathematical Methods of Physics — no further topic split in the official outline.

3. 3

   Linear Algebra

   3 topics

   • Matrices
   • Cayley-Hamilton Theorem
   • Eigenvalues and Eigenvectors
4. 4

   Linear Ordinary Differential Equations

   2 topics

   • First Order
   • Second Order
5. 5

   Special Functions

   4 topics

   • Hermite Functions
   • Bessel Functions
   • Laguerre Functions
   • Legendre Functions
6. 6

   Fourier Series

   overview

   Examined as a single unit within Mathematical Methods of Physics — no further topic split in the official outline.

7. 7

   Fourier and Laplace Transforms

   overview

   Examined as a single unit within Mathematical Methods of Physics — no further topic split in the official outline.

8. 8

   Elements of Complex Analysis

   5 topics

   • Analytic Functions
   • Taylor Series
   • Laurent Series
   • Poles and Residues
   • Evaluation of Integrals
9. 9

   Elementary Probability Theory

   4 topics

   • Random Variables
   • Binomial Distribution
   • Poisson Distribution
   • Normal Distribution
10. 10

    Central Limit Theorem

    overview

    Examined as a single unit within Mathematical Methods of Physics — no further topic split in the official outline.

Mathematical Methods of Physics flashcards for CSIR NET Physical Sciences

22 of 51 cards from the Mathematical Methods of Physics deck — real questions with worked answers.

1. What is a partial differential equation (PDE), and what distinguishes its order and linearity?

   A PDE is an equation involving an unknown function of two or more independent variables and its partial derivatives. Its order is the order of the highest derivative; it is linear if the unknown function and all its derivatives appear to the first power with coefficients depending only on the independent variables, e.g. $a\,u_{xx}+b\,u_{xy}+c\,u_{yy}+\dots=f$.

2. How are second-order linear PDEs of the form $Au_{xx}+Bu_{xy}+Cu_{yy}+\dots=0$ classified?

   By the discriminant $B^{2}-4AC$: elliptic if $B^{2}-4AC<0$ (e.g. Laplace), parabolic if $B^{2}-4AC=0$ (e.g. heat), and hyperbolic if $B^{2}-4AC>0$ (e.g. wave).

3. Write the Laplace equation in three dimensions and state what kind of PDE it is.

   $$\nabla^{2}\phi=\frac{\partial^{2}\phi}{\partial x^{2}}+\frac{\partial^{2}\phi}{\partial y^{2}}+\frac{\partial^{2}\phi}{\partial z^{2}}=0$$ It is a second-order linear elliptic PDE; its solutions are called harmonic functions.

4. What is Poisson's equation and how does it relate to Laplace's equation?

   $$\nabla^{2}\phi=-\rho/\varepsilon_{0}\quad(\text{or }\nabla^{2}\phi=f)$$ It is the inhomogeneous form of Laplace's equation; when the source term $f=0$ it reduces to Laplace's equation $\nabla^{2}\phi=0$.

5. State the mean value property of harmonic functions (solutions of Laplace's equation).

   The value of a harmonic function at a point equals its average over any sphere (or circle in 2D) centered at that point. Consequently harmonic functions attain their maxima and minima only on the boundary (maximum principle).

6. Write the one-dimensional wave equation and give its general d'Alembert solution.

   $$\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial x^{2}}$$ General solution: $u(x,t)=f(x-ct)+g(x+ct)$, a superposition of right- and left-moving waves of speed $c$.

7. Using separation of variables on $u_{tt}=c^{2}u_{xx}$ for a string of length $L$ fixed at both ends, what are the normal-mode solutions?

   $$u_{n}(x,t)=\sin\!\left(\frac{n\pi x}{L}\right)\left[A_{n}\cos\!\frac{n\pi c t}{L}+B_{n}\sin\!\frac{n\pi c t}{L}\right]$$ with allowed frequencies $\omega_{n}=\dfrac{n\pi c}{L}$, $n=1,2,3,\dots$

8. Write the heat (diffusion) equation in two spatial dimensions.

   $$\frac{\partial u}{\partial t}=\alpha\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right)$$ where $\alpha$ is the thermal diffusivity. It is a parabolic PDE.

9. Write the heat equation in three spatial dimensions in compact form.

   $$\frac{\partial u}{\partial t}=\alpha\,\nabla^{2}u=\alpha\left(\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}\right)$$ with thermal diffusivity $\alpha$.

10. For the 1D heat equation by separation of variables, what is the time dependence of each Fourier mode?

    Each spatial mode $\sin(k_{n}x)$ decays exponentially in time as $e^{-\alpha k_{n}^{2}t}$. Higher spatial frequencies decay faster, so the heat equation smooths out the temperature distribution.

11. Contrast the qualitative behavior of solutions to the wave equation versus the heat equation.

    The wave equation (hyperbolic) propagates disturbances at finite speed $c$ without dissipation and is time-reversible. The heat equation (parabolic) diffuses/smooths disturbances, has infinite propagation speed, is dissipative and time-irreversible.

12. In separation of variables for the 2D Laplace equation in Cartesian coordinates, what form do the separated ODEs take?

    Writing $u=X(x)Y(y)$ gives $\dfrac{X''}{X}=-\dfrac{Y''}{Y}=-k^{2}$, so $X''+k^{2}X=0$ (oscillatory) and $Y''-k^{2}Y=0$ (exponential/hyperbolic), or vice versa depending on boundary conditions.

13. What are Dirichlet and Neumann boundary conditions for a PDE?

    Dirichlet: the value of the function $u$ is specified on the boundary. Neumann: the value of the normal derivative $\partial u/\partial n$ is specified on the boundary. (A mix of both is a Robin/mixed condition.)

14. What is the goal of computational (numerical) techniques in physics, and why are they needed?

    They provide approximate numerical solutions to mathematical problems (root finding, integration, differential equations) that cannot be solved analytically in closed form, replacing continuous operations with discrete arithmetic the computer can perform.

15. Distinguish round-off error from truncation error in numerical methods.

    Round-off error arises from the finite precision of representing numbers (limited significant digits). Truncation error arises from approximating an infinite/continuous process by a finite one (e.g. cutting off a Taylor series or using finite step size $h$).

16. State the bisection method for finding a root of $f(x)=0$ and its convergence behavior.

    Given $[a,b]$ with $f(a)f(b)<0$, repeatedly take the midpoint $c=\frac{a+b}{2}$ and keep the subinterval where the sign change persists. It converges linearly; the error is halved each step, so $|x-c_{n}|\le\dfrac{b-a}{2^{n}}$.

17. Write the Newton-Raphson iteration formula for finding a root of $f(x)=0$ and state its order of convergence.

    $$x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}$$ It converges quadratically (order 2) near a simple root, provided $f'(x_{n})\neq0$ and the initial guess is sufficiently close.

18. Write the secant method iteration for a root of $f(x)=0$ and give its order of convergence.

    $$x_{n+1}=x_{n}-f(x_{n})\frac{x_{n}-x_{n-1}}{f(x_{n})-f(x_{n-1})}$$ It needs no derivative and converges superlinearly with order $\approx1.618$ (the golden ratio).

19. Compare the Newton-Raphson and bisection methods for root finding.

    Bisection always converges (if a sign change is bracketed) but only linearly. Newton-Raphson converges quadratically (faster) but requires the derivative, may fail or diverge for poor initial guesses or when $f'\approx0$, and is not guaranteed to stay bracketed.

20. What is interpolation, and how does it differ from extrapolation?

    Interpolation estimates the value of a function at a point lying within the range of known data points. Extrapolation estimates values outside that range. Extrapolation is generally far less reliable because the assumed functional behavior is unverified beyond the data.

21. Write the formula for linear (two-point) interpolation between points $(x_{0},y_{0})$ and $(x_{1},y_{1})$.

    $$y=y_{0}+(x-x_{0})\frac{y_{1}-y_{0}}{x_{1}-x_{0}}$$ This fits a straight line through the two data points.

22. Write the Lagrange interpolation polynomial through $n+1$ points $(x_{i},y_{i})$.

    $$P(x)=\sum_{i=0}^{n}y_{i}\,L_{i}(x),\qquad L_{i}(x)=\prod_{\substack{j=0\\ j\neq i}}^{n}\frac{x-x_{j}}{x_{i}-x_{j}}$$ where each $L_{i}(x_{k})=\delta_{ik}$.

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