CSIR NET Physical Sciences Mathematical Methods of Physics Syllabus

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.

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 $L_{i}(x_{k})=\delta_{ik}$.

What is Newton's forward-difference interpolation formula for equally spaced points?

With $p=\dfrac{x-x_{0}}{h}$: $$P(x)=y_{0}+p\,\Delta y_{0}+\frac{p(p-1)}{2!}\Delta^{2}y_{0}+\frac{p(p-1)(p-2)}{3!}\Delta^{3}y_{0}+\cdots$$ where $\Delta$ is the forward-difference operator.

Write the trapezoidal rule for approximating $\int_{a}^{b}f(x)\,dx$ with $n$ equal subintervals of width $h$.

$$\int_{a}^{b}f(x)\,dx\approx\frac{h}{2}\Big[f_{0}+2(f_{1}+f_{2}+\cdots+f_{n-1})+f_{n}\Big]$$ with $h=\dfrac{b-a}{n}$. It approximates the curve by straight-line segments.

What is the order of the error of the composite trapezoidal rule?

The global error is $O(h^{2})$; explicitly $E=-\dfrac{(b-a)h^{2}}{12}f''(\xi)$. The rule is exact for linear (degree-1) polynomials.

Write Simpson's 1/3 rule for $\int_{a}^{b}f(x)\,dx$ and state its requirement on the number of intervals.

$$\int_{a}^{b}f\,dx\approx\frac{h}{3}\Big[f_{0}+4(f_{1}+f_{3}+\cdots)+2(f_{2}+f_{4}+\cdots)+f_{n}\Big]$$ It requires an even number of subintervals $n$ (odd index coefficients 4, even 2).

State the error order of Simpson's 1/3 rule and which polynomials it integrates exactly.

Global error is $O(h^{4})$, with $E=-\dfrac{(b-a)h^{4}}{180}f^{(4)}(\xi)$. It is exact for polynomials up to degree 3 (cubics), since it fits parabolas through triples of points.

Compare the trapezoidal rule and Simpson's 1/3 rule in accuracy and underlying approximation.

Trapezoidal approximates the integrand by straight lines, error $O(h^{2})$, exact for degree 1. Simpson approximates by parabolas over pairs of intervals, error $O(h^{4})$, exact up to degree 3 — substantially more accurate for the same $h$.

Write Simpson's 3/8 rule and state when it is used.

$$\int_{x_{0}}^{x_{3}}f\,dx\approx\frac{3h}{8}\big[f_{0}+3f_{1}+3f_{2}+f_{3}\big]$$ Used when the number of subintervals is a multiple of 3; it fits a cubic through four points and has error $O(h^{4})$.

What is an initial value problem (IVP) for a first-order ODE?

A problem of the form $\dfrac{dy}{dx}=f(x,y)$ with a prescribed initial condition $y(x_{0})=y_{0}$. The goal of numerical methods is to advance the solution stepwise from $x_{0}$ using step size $h$.

Write Euler's method for solving $y'=f(x,y)$, $y(x_{0})=y_{0}$, and give its accuracy.

$$y_{n+1}=y_{n}+h\,f(x_{n},y_{n})$$ It is a first-order method: local truncation error $O(h^{2})$, global error $O(h)$. It is the simplest explicit one-step method.

Describe the modified Euler (Heun / improved Euler) method.

A predictor-corrector: predict $y^{*}_{n+1}=y_{n}+h f(x_{n},y_{n})$, then correct $$y_{n+1}=y_{n}+\frac{h}{2}\big[f(x_{n},y_{n})+f(x_{n+1},y^{*}_{n+1})\big].$$ It is second-order accurate, global error $O(h^{2})$.

Write the classical fourth-order Runge-Kutta (RK4) update formula.

$$y_{n+1}=y_{n}+\frac{1}{6}(k_{1}+2k_{2}+2k_{3}+k_{4})$$ with $k_{1}=hf(x_{n},y_{n})$, $k_{2}=hf(x_{n}+\tfrac{h}{2},y_{n}+\tfrac{k_{1}}{2})$, $k_{3}=hf(x_{n}+\tfrac{h}{2},y_{n}+\tfrac{k_{2}}{2})$, $k_{4}=hf(x_{n}+h,y_{n}+k_{3})$.

What is the order of accuracy of the classical RK4 method?

RK4 has local truncation error $O(h^{5})$ and global error $O(h^{4})$ — fourth-order accurate, using four function evaluations per step.

Compare Euler's method and RK4 for solving ODEs.

Euler uses one slope evaluation per step, is first-order ($O(h)$ global error), and needs very small $h$ for accuracy. RK4 uses four evaluations per step, is fourth-order ($O(h^{4})$), giving far higher accuracy for moderate $h$ at greater cost per step.

In finite-difference methods, write the central-difference approximations for the first and second derivatives.

$$f'(x)\approx\frac{f(x+h)-f(x-h)}{2h},\qquad f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^{2}}$$ both accurate to $O(h^{2})$.

Write the forward and backward first-difference approximations to $f'(x)$ and give their accuracy.

Forward: $f'(x)\approx\dfrac{f(x+h)-f(x)}{h}$; Backward: $f'(x)\approx\dfrac{f(x)-f(x-h)}{h}$. Both are first-order accurate, $O(h)$, while the central difference is $O(h^{2})$.

How is the 2D Laplace equation discretized by finite differences on a uniform grid?

$$\nabla^{2}u\approx\frac{u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j}}{h^{2}}=0$$ giving the five-point formula $u_{i,j}=\tfrac{1}{4}(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1})$ — each interior value is the average of its four neighbors.

What is a tensor, and how is its rank defined?

A tensor is a multilinear geometric object whose components transform in a definite way under coordinate changes. Its rank (order) is the number of indices: rank 0 = scalar, rank 1 = vector, rank 2 = matrix-like (e.g. stress tensor). A rank-$n$ tensor in $d$ dimensions has $d^{n}$ components.

Give the transformation laws for contravariant and covariant tensor components of rank 1.

Contravariant: $A'^{i}=\dfrac{\partial x'^{i}}{\partial x^{j}}A^{j}$. Covariant: $A'_{i}=\dfrac{\partial x^{j}}{\partial x'^{i}}A_{j}$. Indices upstairs are contravariant, downstairs covariant; repeated upper-lower indices are summed (Einstein convention).

What is the role of the metric tensor $g_{ij}$ in raising and lowering indices?

The metric $g_{ij}$ and its inverse $g^{ij}$ convert between covariant and contravariant components: $A_{i}=g_{ij}A^{j}$ (lowering) and $A^{i}=g^{ij}A_{j}$ (raising). It also defines the line element $ds^{2}=g_{ij}\,dx^{i}dx^{j}$.

Distinguish symmetric and antisymmetric tensors of rank 2.

Symmetric: $T_{ij}=T_{ji}$ (e.g. stress, metric). Antisymmetric: $T_{ij}=-T_{ji}$, which forces diagonal elements to vanish (e.g. electromagnetic field tensor $F_{\mu\nu}$). Any rank-2 tensor decomposes into symmetric and antisymmetric parts.

What is a group in the mathematical sense, and what four axioms define it?

A set $G$ with a binary operation satisfying: (1) closure — $a\cdot b\in G$; (2) associativity — $(a\cdot b)\cdot c=a\cdot(b\cdot c)$; (3) identity — $\exists\,e$ with $e\cdot a=a$; (4) inverse — every $a$ has $a^{-1}$ with $a\cdot a^{-1}=e$.

What distinguishes an Abelian group from a non-Abelian group, and which are Lie groups?

An Abelian group has a commutative operation ($ab=ba$); non-Abelian does not. A Lie group is a continuous group whose elements depend smoothly on parameters (e.g. $SU(2)$, $O(3)$), as opposed to a discrete/finite group.

What is a group representation?

A representation is a homomorphism from a group $G$ to a set of linear operators (matrices) on a vector space, $D:G\to GL(V)$, preserving the group multiplication: $D(g_{1})D(g_{2})=D(g_{1}g_{2})$. An irreducible representation has no nontrivial invariant subspace.

Define the group $SU(2)$ and state its dimension.

$SU(2)$ is the group of $2\times2$ complex unitary matrices with determinant $+1$: $U^{\dagger}U=I$, $\det U=1$. It is a 3-parameter (3-dimensional) compact Lie group, the group of spin rotations.

What are the generators of $SU(2)$ and their commutation relations?

The generators are $J_{i}=\tfrac{1}{2}\sigma_{i}$ (half the Pauli matrices), satisfying the angular-momentum Lie algebra $$[J_{i},J_{j}]=i\,\epsilon_{ijk}J_{k}.$$ A general element is $U=e^{-i\theta\,\hat{n}\cdot\vec{J}}$.

Define the group $O(3)$ and distinguish it from $SO(3)$.

$O(3)$ is the group of real $3\times3$ orthogonal matrices ($R^{T}R=I$), with $\det R=\pm1$. $SO(3)$ is the subgroup with $\det R=+1$ (proper rotations). $O(3)$ additionally includes improper operations (reflections/inversion, $\det R=-1$).

What is the relationship (homomorphism) between $SU(2)$ and $SO(3)$?

There is a 2-to-1 homomorphism from $SU(2)$ onto $SO(3)$: each rotation in $SO(3)$ corresponds to two elements $\pm U$ in $SU(2)$. $SU(2)$ is the double cover of $SO(3)$, which is why spin-1/2 states acquire a sign change under a $2\pi$ rotation.

What are the Pauli matrices, and what algebraic identity do they satisfy?

$$\sigma_{1}=\begin{pmatrix}0&1\\1&0\end{pmatrix},\ \sigma_{2}=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\ \sigma_{3}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$ They satisfy $\sigma_{i}\sigma_{j}=\delta_{ij}I+i\epsilon_{ijk}\sigma_{k}$, are Hermitian, traceless, and unitary with $\sigma_{i}^{2}=I$.

For a string fixed at both ends, how is the initial-shape coefficient $B_n$ obtained, and what role does Fourier analysis play in solving the wave/heat equations?

The initial condition is expanded in the orthogonal eigenfunctions $\sin(n\pi x/L)$, giving the Fourier sine coefficients $$B_{n}=\frac{2}{L}\int_{0}^{L}u(x,0)\sin\!\frac{n\pi x}{L}\,dx.$$ Fourier analysis decomposes the boundary/initial data into the normal modes whose time evolution is known, then superposes them.