Mathematical Methods of Physics Vector calculus - Complex analysis - ODE/PDE - Tensors Class notes focused on CSIR-UGC NET/JRF/Ph.D. Part-B & Part-C

 Mathematical Methods of Physics

Vector calculus - Complex analysis - ODE/PDE - Tensors
Class notes focused on CSIR-UGC NET/JRF/Ph.D. Part-B & Part-C

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1. Vector calculus ― one–page essentials

Tool	Key formulas	Quick tip
Gradient	$\nabla f = \hat{i}\partial_x f + \hat{j}\partial_y f + \hat{k}\partial_z f$	Direction of steepest rise
Divergence	$\nabla\!\cdot\!\mathbf{A}=\partial_x A_x+\partial_y A_y+\partial_z A_z$	>0 ⇒ source; <0 ⇒ sink
Curl	$\nabla\!\times\!\mathbf{A}= \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\ \partial_x&\partial_y&\partial_z\\ A_x&A_y&A_z\end{vmatrix}$	Non-zero ⇒ field rotational
Integral theorems	$\displaystyle\oint \mathbf{A}\!\cdot\!d\mathbf{l}= \iint (\nabla\!\times\!\mathbf{A})\!\cdot\!d\mathbf{S}$ (Stokes)
$\displaystyle\iint \mathbf{A}\!\cdot\!d\mathbf{S}= \iiint (\nabla\!\cdot\!\mathbf{A})\,dV$ (Gauss)	Always sketch surface orientation first

Curvilinear coordinates: scale factors $h_i$;
$\nabla f = \sum_i \hat{e}_i\frac{1}{h_i}\partial_{q_i}f$.
Remember $h_r=1,\;h_\theta=r,\;h_\phi=r\sin\theta$ in spherical.

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2. Complex analysis ― NET-level core

1. Cauchy–Riemann (Cartesian)
   $u_x=v_y,\;u_y=-v_x$ ⇒ analytic at point.

2. Cauchy integral formula
   $f^{(n)}(z_0)=\dfrac{n!}{2\pi i}\displaystyle\oint_{C}\dfrac{f(z)}{(z-z_0)^{n+1}}dz$.

3. Maximum-modulus principle
   |f(z)| attains max on boundary of domain.

4. Residue at simple pole $z_0$
   $\text{Res}[f,z_0]=\displaystyle\lim_{z\to z_0}(z-z_0)f(z)$.
   Net contour integral $=2\pi i\sum\text{Residues}$.

5. Quick Laurent check
   If pole order m, principal part has m terms only.

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3. Ordinary differential equations (ODE)

Class	Canonical form	NET favourite
2nd-order linear	$y''+P(x)y'+Q(x)y=R(x)$	Frobenius at regular singular point
Sturm–Liouville	$(p y')'+\lambda w y=0$	Orthogonality: $\int w\,y_m y_n=0\;(m\ne n)$
Green’s function	$Ly=f\;\Rightarrow\;y=\int G(x,\xi)f(\xi)d\xi$	Write G from two independent solutions

Variation-of-parameters recipe
$y=y_1\int\!\dfrac{y_2 R}{W}\,dx - y_2\int\!\dfrac{y_1 R}{W}\,dx$.

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4. Partial differential equations (PDE)

1. Classification (2-D) by $B^2-4AC$ for $A u_{xx}+B u_{xy}+C u_{yy}$:

   • <0 elliptic (Laplace) - =0 parabolic (heat) - >0 hyperbolic (wave).

2. Separation of variables quick template (Cartesian)
   Assume $u(x,t)=X(x)T(t)$ ⇒ split into two ODEs with separation constant −λ.

3. First-order PDE – Lagrange auxiliary system
   For $P\,p+Q\,q = R$ use $\dfrac{dx}{P}=\dfrac{dy}{Q}=\dfrac{dz}{R}$.

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5. Tensors & curvilinear coordinates

• Tensor of rank r: transforms with r direction cosines.

• Metric tensor $g_{ij}= \mathbf{e}_i\!\cdot\!\mathbf{e}_j$; in spherical $g_{ij}=\text{diag}(1,r^{2},r^{2}\sin^{2}\theta)$.

• Christoffel symbols
  $\Gamma^{k}_{ij}= \tfrac12 g^{kl}(\partial_i g_{lj} + \partial_j g_{il} - \partial_l g_{ij})$.

• Covariant derivative $A^{i}{}_{;j}= \partial_j A^{i}+ \Gamma^{i}_{kj}A^{k}$.

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6. Speed-drill examples

1. Residue evaluation
   ∮ $\dfrac{e^{iz}}{z^{2}(z-1)}dz$ around |z|=½.
   Only pole at z=0 inside. Order 2 ⇒ residue $=\frac{1}{1!}\,\partial_z[e^{iz}/(z-1)]_{z=0}= -1$.
   Integral = $2\pi i(-1)=-2\pi i$.

2. Green’s function for $y''=f(x)$ on 0<x<1, y(0)=y(1)=0.
   $G(x,\xi)=\begin{cases} x(1-\xi), & x<\xi\\ \xi(1-x), & x>\xi \end{cases}$.

3. Tensor contraction
   In 3-D, the double dot of two second-rank tensors $A:B = A_{ij}B_{ij}$.

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7. 15-day micro-plan

• Days 1-3: Vector calculus identities & integral theorems—prove them once, then derive EM problems.

• Days 4-6: Complex analysis—daily 20 residue questions; memorise standard series (ln(1+z), sin z).

• Days 7-10: ODE/PDE—solve one Sturm–Liouville and one separation-of-variables set each day.

• Days 11-13: Tensors—rewrite Maxwell eqns in spherical; practice raising/lowering indices.

• Days 14-15: Mixed past CSIR problems; simulate 3-hr Part-B/Part-C.