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GATE Engineering Mathematics Core Formulas

Linear algebra, vector calculus, differential equations, and numerical methods formulas for GATE engineering preparation.

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Sum and Product of Eigenvalues

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Sum and Product of Eigenvalues

Sum of eigenvalues equals

trace (A)

; product of eigenvalues equals

det (A)

Cayley-Hamilton Theorem

Every square matrix satisfies its own characteristic polynomial equation:

p (A) = 0

Rank-Nullity Theorem

Rank (T) + Nullity (T) = n

, where

n

is dimension of domain space.

Orthogonal Matrix Property

A^{T} A = A A^{T} = I

, implying

A^{- 1} = A^{T}

and

det (A) = \pm 1

Divergence of Vector Field

F

\nabla \cdot F = \frac{\partial F _{x}}{\partial x} + \frac{\partial F _{y}}{\partial y} + \frac{\partial F _{z}}{\partial z}

, yielding a scalar field.

Curl of Conservative Vector Field

\nabla \times F = 0

, meaning line integrals are path independent.

Green's Theorem in the Plane

\oint_{C} (P d x + Q d y) = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A

Stokes' Theorem Formulation

\oint_{C} F \cdot d r = \iint_{S} (\nabla \times F) \cdot d S

Gauss Divergence Theorem

\iint_{S} F \cdot d S = ∭_{V} (\nabla \cdot F) d V

Integrating Factor for Linear First-Order ODE

For

y^{'} + P (x) y = Q (x)

, integrating factor is

I (x) = e^{\int P (x) d x}

Cauchy-Euler Equation Substitution

Substitute

x = e^{z}

(

z = ln x

), transforming

x^{2} y^{''}

into

D (D - 1) y

Cauchy-Riemann Equations in Cartesian Form

u_{x} = v_{y}

and

u_{y} = - v_{x}

, necessary conditions for analyticity.

Harmonic Function Property

Real and imaginary parts of an analytic function satisfy Laplace's equation:

\nabla^{2} u = 0

Cauchy Residue Theorem

\oint_{C} f (z) d z = 2 π i \sum Res (f, z_{k})

for singularities inside

C

Laplace Transform of

e^{a t} sin (ω t)

L {e^{a t} sin (ω t)} = \frac{ω}{( s - a ) ^{2} + ω ^{2}}

for

s > a

Final Value Theorem in Laplace Domain

lim_{t \to \infty} f (t) = lim_{s \to 0} s F (s)

, provided poles lie in open LHP.

Fourier Series

a_{0}

Coefficient (Period

2 L

)

a_{0} = \frac{1}{L} \int_{- L}^{L} f (x) d x

, representing twice the average value.

Newton-Raphson Iteration Formula

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

, exhibiting quadratic convergence near roots.

Simpson's 1/3 Rule Step Requirement

Requires an even number of subintervals

n

(

h = (b - a) / n

) and has error order

O (h^{4})

Trapezoidal Rule Formulation

\int_{a}^{b} f (x) d x \approx \frac{h}{2} [f_{0} + 2 (f_{1} + \dots + f_{n - 1}) + f_{n}]

Poisson Distribution PMF

P (X = k) = \frac{λ ^{k} e ^{- λ}}{k !}

, with mean

=

variance

= λ

Normal Distribution 68-95-99.7 Rule

68.27%

within

\pm 1 σ

95.45%

within

\pm 2 σ

99.73%

within

\pm 3 σ

Variance Formula via Expectations

Var (X) = E [X^{2}] - (E [X])^{2}

, always non-negative.

Directional Derivative and Gradient

D_{u} f = \nabla f \cdot \overset{u}{^}

, maximized in the direction of

\nabla f

Wronskian Test for Linear Independence

W (y_{1}, y_{2}) (x) \neq = 0

at some point in interval, solutions are linearly independent.

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