Engineering Mathematics Syllabus

April 02, 2025

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Engineering Mathematics

Complete Guide for GATE Metallurgy (MT) - Section 1 (TestUrSelf)

Table of Contents

1. Linear Algebra
2. Calculus
3. Vector Calculus
4. Differential Equations
5. Probability & Statistics
6. Numerical Methods

1.1 Linear Algebra

Matrices & Determinants

A

Matrix Operations

|A|

Determinants

A⁻¹

Inverse Matrix

Matrix Example

a₁₁

a₁₂

a₁₃

a₂₁

a₂₂

a₂₃

a₃₁

a₃₂

a₃₃

det(A) = a₁₁(a₂₂a₃₃ - a₂₃a₃₂) - a₁₂(a₂₁a₃₃ - a₂₃a₃₁) + a₁₃(a₂₁a₃₂ - a₂₂a₃₁)

Eigenvalues & Eigenvectors

Ax = λx

Where:

• A = square matrix
• λ = eigenvalue (scalar)
• x = eigenvector (non-zero)

Example: Find Eigenvalues

For matrix A = [[2, 1], [1, 2]]:

det(A - λI) = (2-λ)² - 1 = 0 → λ = 1, 3

1.2 Calculus

Limits & Continuity

lim_x→a f(x) = L if ∀ε>0, ∃δ>0 : 0 < |x-a| < δ ⇒ |f(x)-L| < ε

Key Concepts

• Continuity: lim_x→a f(x) = f(a)
• Differentiability: f'(a) exists
• L'Hôpital's Rule for 0/0 or ∞/∞ forms

Maxima & Minima

Critical points: f'(x) = 0 or undefined

Second derivative test: f''(x) > 0 ⇒ local min, f''(x) < 0 ⇒ local max

1.3 Vector Calculus

Gradient, Divergence & Curl

∇f

Gradient

∇·F

Divergence

∇×F

Curl

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

∇·F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z

∇×F = (∂F_z/∂y - ∂F_y/∂z, ∂F_x/∂z - ∂F_z/∂x, ∂F_y/∂x - ∂F_x/∂y)

Integral Theorems

Theorem	Equation	Application
Green's	∮C (Pdx + Qdy) = ∬D (∂Q/∂x - ∂P/∂y) dA	2D regions
Stokes'	∮C F·dr = ∬S (∇×F)·dS	3D surfaces
Gauss'	∯S F·dS = ∭V (∇·F) dV	3D volumes

1.4 Differential Equations

Ordinary Differential Equations

First Order ODEs

dy/dx + P(x)y = Q(x) (Linear)

M(x,y)dx + N(x,y)dy = 0 (Exact)

Second Order Linear

ay'' + by' + cy = 0 → Characteristic equation: ar² + br + c = 0

Partial Differential Equations

Equation	Form	Solution Method
Laplace	∇²u = 0	Separation of variables
Heat	∂u/∂t = α∇²u	Fourier series
Wave	∂²u/∂t² = c²∇²u	d'Alembert's solution

1.5 Probability & Statistics

Probability Basics

P(A∪B) = P(A) + P(B) - P(A∩B)

P(A|B) = P(A∩B)/P(B) (Conditional Probability)

μ

Mean

σ

Std Dev

Probability Distributions

Distribution	PMF/PDF	Parameters
Binomial	C(n,k)pk(1-p)n-k	n, p
Poisson	(λke-λ)/k!	λ
Normal	(1/σ√2π)e-(x-μ)²/2σ²	μ, σ

1.6 Numerical Methods

Root Finding Methods

B

Bisection

N

Newton-Raphson

S

Secant

Newton-Raphson: x_n+1 = x_n - f(x_n)/f'(x_n)

Numerical Integration

Trapezoidal: ∫_a^b f(x)dx ≈ (h/2)[f(x₀) + 2Σf(xᵢ) + f(xₙ)]

Simpson's: ∫_a^b f(x)dx ≈ (h/3)[f(x₀) + 4Σf(x_odd) + 2Σf(x_even) + f(xₙ)]

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