High-Yield Math Cheatsheet

Interactive Formula & Theorem Bank

Master the most critical mathematical formulas, theorems, and definitions tested in CSIR NET, GATE, and IIT-JAM with complete LaTeX rendering.

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Real AnalysisHigh Yield for IIT-JAM & CSIR NET Part B/C

Weierstrass M-Test for Uniform Convergence

$$\text{If } |f_n(x)| \le M_n \quad \forall x \in E \quad \text{and} \quad \sum_{n=1}^\infty M_n < \infty, \quad \text{then } \sum_{n=1}^\infty f_n(x) \text{ converges uniformly on } E.$$

A primary test used in CSIR NET & IIT-JAM to establish the uniform convergence of series of functions.

#series#uniform convergence#analysis

Real AnalysisEssential Conceptual Question in CSIR NET Part C

Riemann Integrability Criterion (Darboux)

$$f \in \mathcal{R}[a,b] \iff \forall \varepsilon > 0, \; \exists \text{ partition } P \text{ such that } U(P, f) - L(P, f) < \varepsilon$$

Necessary and sufficient condition for a bounded function to be Riemann integrable on a closed interval [a, b].

#riemann#integration#darboux

Real AnalysisDirect application in GATE MA & IIT-JAM

Taylor's Theorem with Lagrange Form of Remainder

$$f(x) = \sum_{k=0}^{n-1} \frac{f^{(k)}(a)}{k!}(x-a)^k + \frac{f^{(n)}(\xi)}{n!}(x-a)^n \quad \text{for some } \xi \in (a,x)$$

Approximation theorem with explicit remainder term used in error bounds and calculus questions.

#taylor#calculus#approximation

Linear AlgebraGuaranteed 1-2 questions in every CSIR NET & GATE paper

Cayley-Hamilton Theorem & Minimal Polynomial

$$\chi_A(\lambda) = \det(\lambda I - A) \implies \chi_A(A) = 0 \quad \text{and} \quad m_A(\lambda) \mid \chi_A(\lambda)$$

Every square matrix satisfies its own characteristic equation, and the minimal polynomial divides the characteristic polynomial.

#matrix#eigenvalues#minimal polynomial

Linear AlgebraTop Tested Concept in CSIR NET Part B & C

Matrix Diagonalizability Criterion

$$A \in M_n(\mathbb{F}) \text{ is diagonalizable} \iff m_A(x) = (x-\lambda_1)(x-\lambda_2)\cdots(x-\lambda_k) \quad (\text{distinct linear factors})$$

A linear operator is diagonalizable if and only if its minimal polynomial factors into distinct linear roots over the field.

#diagonalization#eigenvectors#jordan

Linear AlgebraUniversal Foundation for All Exam Linear Algebra

Dimension / Rank-Nullity Theorem

$$\dim(V) = \operatorname{rank}(T) + \operatorname{nullity}(T) = \dim(\operatorname{Im} T) + \dim(\ker T)$$

Fundamental dimensional relationship between domain, kernel, and image of any linear transformation.

#rank#nullity#dimension

Complex AnalysisGuaranteed 4-8 marks in GATE & CSIR NET

Cauchy's Residue Theorem

$$\oint_{\gamma} f(z)\, dz = 2\pi i \sum_{k=1}^m \operatorname{Res}(f, z_k) \quad \text{where } \operatorname{Res}(f, z_0) = \lim_{z \to z_0} \frac{1}{(n-1)!}\frac{d^{n-1}}{dz^{n-1}}\left[(z-z_0)^n f(z)\right]$$

Powerful contour integral evaluation formula summing residues of singularities enclosed inside contour gamma.

#contour integral#residues#singularities

Complex AnalysisCore Requirement for IIT-JAM & CSIR NET

Cauchy-Riemann Equations (Polar & Cartesian)

$$u_x = v_y, \quad u_y = -v_x \qquad \text{and in polar: } \quad u_r = \frac{1}{r}v_\theta, \quad v_r = -\frac{1}{r}u_\theta$$

Necessary and sufficient conditions for complex differentiability / analyticity of f(z) = u(x,y) + i*v(x,y).

#analytic functions#harmonic#cauchy-riemann

Complex AnalysisFrequently asked in CSIR NET Part C & GATE

Rouché's Theorem on Zeroes of Analytic Functions

$$\text{If } |f(z)| > |g(z)| \; \forall z \in \gamma, \quad \text{then } f(z) \text{ and } f(z) + g(z) \text{ have same number of zeroes inside } \gamma.$$

Used to count the number of roots of complex polynomials inside circles or specific regions without factoring.

#roots#zeroes#contour

Differential EquationsStandard Question in IIT-JAM, GATE & CSIR NET

Wronskian & Abel's Identity for Second Order ODE

$$y'' + P(x)y' + Q(x)y = 0 \implies W(x) = W(x_0) \exp\left(-\int_{x_0}^x P(t)\, dt\right)$$

Calculates the Wronskian determinant W(y1, y2) directly without explicitly solving the differential equation.

#wronskian#ode#linear independence

Differential EquationsKey Applied Math Topic in CSIR NET & GATE

d'Alembert's Formula for 1D Wave Equation

$$u_{tt} = c^2 u_{xx} \implies u(x,t) = \frac{f(x-ct) + f(x+ct)}{2} + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\, ds$$

Exact analytical solution of the homogeneous one-dimensional wave equation with initial position f(x) and velocity g(x).

#pde#wave equation#dalembert

Modern AlgebraFundamental in CSIR NET & GATE MA

Class Equation of a Finite Group

$$|G| = |Z(G)| + \sum_{i=1}^k [G : C_G(x_i)] = |Z(G)| + \sum_{i=1}^k \frac{|G|}{|C_G(x_i)|}$$

Partitions a finite group into the center Z(G) and non-central conjugacy classes. Central to proving p-group properties.

#groups#conjugacy#sylow

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