Engineering Mathematics
“Master Linear Algebra, Multivariable Calculus, and Probability Distributions — the formal mathematical backbone of Computer Science and GATE.”
A rigorous, mathematically formal study notebook covering Linear Algebra, Differential and Integral Calculus, and Probability Theory tailored specifically for undergraduate computer science and GATE CS/IT high-scorers.
Structured Notes & Exam Lens
A · B.Tech Structured Study Sequence
5 Sections · Full CurriculumMatrices, Linear Systems & LU Factorization
Matrix algebra, determinants, rank of a matrix, Gaussian elimination, consistency of linear systems AX = B, and LU decomposition.
Echelon forms, Rouché-Capelli theorem, unique vs infinite solutions, and Doolittle LU factorization.
Characteristic polynomials, eigenspaces, algebraic vs geometric multiplicity, and matrix powers.
Differential Calculus & Mean Value Theorems
Limits, indeterminate forms, continuity, differentiability, Rolle's, Lagrange's, Cauchy's Mean Value Theorems, and Taylor series.
L'Hôpital's rule, 1^infinity exponential identities, left/right limits, and smoothness conditions.
Rolle's, Lagrange's, Cauchy's MVT, Taylor polynomial approximations, and remainder formulas.
Maxima-Minima & Definite/Improper Integrals
Critical points, single and multivariable extrema, Hessian discriminant test, definite integrals, and improper integral convergence.
First/second derivative tests, Hessian matrix determinant test D = f_xx * f_yy - (f_xy)^2.
Properties of definite integrals, Leibniz integral rule, Gamma function Gamma(n), and improper convergence.
Probability Foundations & Random Variables
Sample spaces, conditional probability, Bayes' Theorem, discrete/continuous random variables, expectation, variance, and covariance.
Prior to posterior updates, base-rate fallacy, partition of sample spaces, and conditional independence.
PDF/CDF properties, Linearity of Expectation, Covariance, and Chebyshev's inequality.
Standard Distributions & Descriptive Statistics
Binomial, Poisson, Uniform, Exponential, Normal distributions, Central Limit Theorem, and descriptive metrics (mean, median, mode, variance).
Binomial, Poisson approximation, Uniform, Exponential memoryless property, and Normal / Gaussian curve.
Mean, median, mode, standard deviation, skewness, sample mean variance, and the Central Limit Theorem.
B · GATE 2027 Syllabus Scope
Official syllabus scope and related lesson coverage; topics outside the syllabus are marked as enrichment.
- •Linear Algebra: Matrices, determinants, system of linear equations, eigenvalues and eigenvectors, LU decomposition
- •Calculus: Limits, continuity and differentiability, max and min, mean value theorem, evaluation of definite and improper integrals
- •Probability and Statistics: Random variables, uniform, normal, exponential, Poisson and binomial distributions, mean, median, mode and standard deviation, conditional probability and Bayes theorem
Interactive Labs & Verified PYQs
Revision & Rapid Review Sheets
Authoritative Textbooks & Video Lectures
All notes in this notebook cross-reference standard academic literature. We do not pretend student notes substitute for classical texts — they guide you through them.
Curated lectures from verified educators (Gate Smashers, Neso Academy, Knowledge Gate, and MIT OCW) selected specifically for difficult concept visualizations.
These notes are open and maintained together.
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