CSIR NET Physical Sciences Mathematical Methods of Physics Flashcards

24 sample cards from the Mathematical Methods of Physics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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 $h$.

2. 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 $O(h^{4})$.

3. 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 $h$.

4. 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.

5. 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 $O(h^{2})$.

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

7. 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.

8. 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.

9. 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 $O(h^{2})$.

10. 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 $O(h^{2})$.

11. 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.

12. 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.

13. 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).

14. 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 $ds^{2}=g_{ij}\,dx^{i}dx^{j}$.

15. 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.

16. 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 $a\cdot a^{-1}=e$.

17. 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.

18. 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.

19. 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.

20. 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 $U=e^{-i\theta\,\hat{n}\cdot\vec{J}}$.

21. 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, $\det R=-1$).

22. 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.

23. 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 $\sigma_{i}^{2}=I$.

24. 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.

What this deck covers

Mathematical Methods of Physics flashcards FAQ