GATE Mathematics — Syllabus, Topics & Key Formulas

GATE Mathematics usually means the MA paper, a broad syllabus covering 11 major areas from Calculus and Linear Algebra to Topology and Linear Programming, plus the General Aptitude section common to every GATE paper. The single most important thing to get right first is the label: the MA paper is not the Engineering Mathematics portion inside papers such as EE, ME, or CSE. Mix those up and your study plan is wrong before you start.

When this overview is most useful

Lean on this page at two moments: at the start of preparation, when you need to see the shape of the MA paper quickly, and during revision, when you decide what belongs on a formula sheet versus a theorem sheet. It is also useful if you are switching from a branch-specific GATE paper to MA, because the jump is not only about difficulty — it is about the type of mathematics being tested.

The approach, step by step

1. Confirm the paper. Decide whether you mean the GATE Mathematics paper (MA) or the Engineering Mathematics part of another paper; the scope differs sharply.
2. Map the units. The MA syllabus groups into these broad blocks: Calculus, Linear Algebra, Real Analysis, Complex Analysis, Ordinary Differential Equations, Algebra, Functional Analysis, Numerical Analysis, Partial Differential Equations, Topology, and Linear Programming.
3. Separate formulas from theory. Sort units by how you actually study them:
   • Calculation-heavy: Calculus, Linear Algebra, Complex Analysis, ODEs, Numerical Analysis, PDEs, parts of Linear Programming.
   • Definition-and-theorem-heavy: Real Analysis, Algebra, Functional Analysis, Topology.
   • Mixed (both technique and theory): Complex Analysis, Linear Algebra. Keep a formula sheet for the computational blocks and a separate condition sheet for the proof-heavy ones.
4. Practice one core method per unit — Newton-Raphson, eigenvalue computation, residue calculation, simplex setup — so you can execute at least one standard technique cleanly.
5. Check conditions before using any formula or theorem. Many wrong solutions look plausible because the theorem is correct but its hypotheses were never verified.

If you remember one idea, make it this: GATE Mathematics is not a single "formula subject." Some units reward fast computation; others reward careful use of definitions and hypotheses.

The phrase "key formulas" is useful only up to a point. In several MA topics the real scoring difference comes from knowing when a theorem applies, not from memorizing a long list. In Real Analysis, the dominated convergence theorem is powerful, but only when its hypotheses are genuinely satisfied. In Algebra, knowing the statement of Sylow's theorems matters more than carrying a formula sheet. In Topology, definitions such as compactness, connectedness, basis, and quotient topology do most of the work. So the practical rule is simple: keep a formula sheet for the computational blocks, and a separate condition sheet for the proof-heavy ones.

Anchor formulas worth revising

These are recognition anchors, not the whole syllabus.

Calculus and optimization. For $f(x_1,\dots,x_n)$ the gradient is

and for a smooth constraint $g = c$, the Lagrange condition $\nabla f = \lambda \nabla g$ holds at candidate points — it does not by itself guarantee a max or min.

Linear algebra. The characteristic polynomial $p_A(\lambda) = \det(\lambda I - A)$; eigenpairs satisfy $Av = \lambda v$; and rank-nullity gives $\dim(V) = \operatorname{rank}(T) + \operatorname{nullity}(T)$.

Complex analysis. With $f$ analytic on and inside a suitable contour $C$, Cauchy's formula $f(a) = \tfrac{1}{2\pi i}\oint_C \tfrac{f(z)}{z-a}\,dz$, and the residue theorem $\oint_C f(z)\,dz = 2\pi i \sum \operatorname{Res}(f; a_k)$ — both need the analyticity and contour assumptions in place.

Numerical analysis. Newton-Raphson $x_{n+1} = x_n - \tfrac{f(x_n)}{f'(x_n)}$ (requires $f'(x_n) \ne 0$ and a reasonable start), and the composite trapezoidal rule with $h = \tfrac{b-a}{n}$:

an approximation, not an identity.

Differential equations and transforms. The Laplace transform $\mathcal{L}\{f\}(s) = \int_0^\infty e^{-st} f(t)\,dt$ when it converges; in PDE, the workflow (classification, canonical forms, separation, transforms) matters more than any one formula.

Linear programming. A standard model: maximize or minimize $c^T x$ subject to $Ax \le b$, $x \ge 0$. Setup is as important as solving.

Full example: Newton-Raphson for $\sqrt{2}$

Take $f(x) = x^2 - 2$, so $f'(x) = 2x$ and

Start at $x_0 = 1.5$:

already very close to $\sqrt{2} \approx 1.4142$. This is the difference between knowing a formula and knowing a method: many MA questions are really about turning one standard method into a clean sequence of steps.

Where preparation goes wrong

• Mixing MA with Engineering Mathematics — the most basic error; MA is much broader.
• Building only a formula sheet — useless for Real Analysis, Algebra, Functional Analysis, Topology, where definitions and theorem conditions carry the weight.
• Using a theorem without its hypotheses — common with convergence theorems, inverse/implicit function theorems, and contour-integration results.
• Treating numerical methods as exact — Newton-Raphson, trapezoidal, Simpson, Jacobi, Gauss-Seidel all carry approximation or convergence conditions.
• Ignoring problem setup in Linear Programming — the real mistake is usually a wrong objective or constraint, one step before the algebra.

GATE Mathematics behaves like a formula subject in units where you repeatedly apply a standard tool (eigenvalues, residues, Newton-Raphson, solving an ODE, setting up an LP) and less so where definitions and hypotheses must be interpreted carefully.

A revision plan that matches the syllabus

Split your notes into three compact sections: one for formulas and computational templates, one for definitions, theorem statements, and standard counterexamples, and one for short solved problems. That structure matches the syllabus far better than a single long notebook, and it keeps you from over-studying formula-heavy units while under-studying proof-heavy ones. As practice, pick one MA unit and make a one-page summary with two parts — the conditions you must not forget, and the two or three formulas or methods you use most in that unit. If you want to check the arithmetic inside a step such as a Newton-Raphson iteration or an eigenvalue calculation, redo it by hand and compare against your worked notes before trusting it.