01 · Foundation
Algebra, Functions, and Engineering Notation
Rearranging any relationship and tracking units: the price of admission to every later chapter.
Start chapter →Foundations · Built on Kreyszig, Advanced Engineering Mathematics
Mathematics for Mechanical Engineers
The advanced core of this course follows one main text, Kreyszig's Advanced Engineering Mathematics. Because that book begins at differential equations, the algebra, trigonometry, and calculus it assumes are taught here in full, with their own recommended texts. The goal is engineering fluency, not mathematical perfection.
01
How this course is designed
Tool-first, not proof-first
Every chapter exists because a later engineering course needs it: Statics, Dynamics, Mechanics of Materials, Thermodynamics, Fluids, Heat Transfer, Controls, Vibrations, FEA, and Numerical Methods. Each chapter states its engineering connection up front.
Same learning system as every course
Readiness check with routing, core idea with limits, a fully worked example with real numbers and a figure, misconception table, four-level practice ladder, retrieval quiz with spaced review, AI guidance, and a portfolio task.
One main text for the core
The advanced chapters map directly to Kreyszig, Advanced Engineering Mathematics. Series, complex analysis, PDEs, and advanced optimization are named where they fit the engineering path. The foundations use standard calculus references, listed below.
02
The course map
Nineteen chapters, one story. Each act answers a question the last one raised, so the maths arrives exactly when an engineering problem needs it.
Act 1 · The language: how to describe things
Foundation, learn it fully here. School-level, and everything later rests on it.
02 · Foundation
Trigonometry and Geometry for Mechanics
Triangles, angles, and the shape of oscillation: forces resolve and motion repeats through these.
Start chapter →03 · Foundation
Vectors and Coordinate Systems
Quantities with direction, built from the triangles before them: forces, velocities, and moments.
Start chapter →Act 2 · How things change: calculus
Foundation, learn it fully here. This is first-year calculus.
04 · Foundation
Single-Variable Calculus: Derivatives
The rate at which things change: velocity from position, slope from a curve, sensitivity from a formula.
Start chapter →05 · Foundation
Single-Variable Calculus: Integrals
Adding up change: distance from velocity, work from force, the resultant of a distributed load.
Start chapter →06 · Stewart, series
Sequences, Series, and Taylor Approximation
Replacing a hard function with a short polynomial: why small-angle works and how calculators compute.
Start chapter →Act 3 · Many things at once
Core, learn the ideas here, then build real fluency with the textbook problem sets.
07 · Kreyszig calculus
Multivariable Calculus
When a quantity depends on several inputs: partial derivatives, the gradient, and multiple integrals.
Start chapter →08 · Kreyszig Ch 9 to 10
Vector Calculus for Mechanical Engineers
Fields that fill space: how heat flows and fluids spread, through gradient, divergence, and curl.
Start chapter →Act 4 · Many equations at once: linear algebra
Core, learn the ideas here, then build real fluency with the textbook problem sets.
09 · Kreyszig Ch 7
Matrices and Systems of Linear Equations
Handling dozens of equations together: trusses, circuits, and the systems FEA assembles.
Start chapter →10 · Kreyszig Ch 8
Eigenvalues, Eigenvectors, and Modes
The natural directions and frequencies hidden in a system: vibration modes and stability.
Start chapter →Act 5 · Change over time: the heart of mechanical engineering
Advanced, meet each idea here so it makes sense, then master it with the textbook and the engineering courses that use it.
11 · Kreyszig Ch 13
Complex Numbers for Engineers
The language of rotation and oscillation, picking up the complex roots the earlier chapters left waiting.
Start chapter →12 · Kreyszig Ch 1 to 3
Ordinary Differential Equations
The master equation of mechanical systems: mass, damper, spring, and everything that settles or rings.
Start chapter →13 · Kreyszig Ch 4
Systems of ODEs and State-Space Thinking
Many coupled states at once, their fate read straight from the eigenvalues of Act 4.
Start chapter →14 · Kreyszig Ch 6
Laplace Transforms and Transfer Functions
Turning differential equations into algebra: the native language of control systems.
Start chapter →15 · Kreyszig Ch 11
Fourier Series, Frequency, and Signals
Every signal as a sum of sines: vibration spectra, resonance, and what an FFT shows.
Start chapter →16 · Kreyszig Ch 12
Partial Differential Equations
Change across space and time together: the heat, wave, and Laplace equations behind fields.
Start chapter →Act 6 · Meeting the real world
Applied, learn the method here and deepen it with real tools and data.
17 · Kreyszig Ch 19 to 21
Numerical Methods for Mechanical Engineers
When there is no formula: the honest approximations inside every simulation you will run.
Start chapter →18 · Kreyszig Ch 24 to 25
Probability, Statistics, and Engineering Uncertainty
Real measurements scatter and parts vary: making defensible decisions under doubt.
Start chapter →19 · Kreyszig Ch 22
Engineering Optimization
Finding the best design within its constraints: gradients to zero, Lagrange multipliers, and gradient descent.
Start chapter →03
Textbooks and what is missing
The advanced core runs on one main text. The foundations use standard calculus and precalculus references, since Kreyszig does not cover them. Pick one option per row.
| Part of the course | Course chapters | Main text |
|---|---|---|
| Algebra, trigonometry, functions | 1 to 3 | OpenStax Precalculus (free), or Stewart, Redlin and Watson, Precalculus |
| Single-variable calculus and series | 4 to 6 | OpenStax Calculus, volumes 1 to 2 (free), or Stewart, Calculus: Early Transcendentals |
| Multivariable and vector calculus | 7 to 8 | Stewart, Calculus (multivariable), then Kreyszig, Advanced Engineering Mathematics, Ch 9 to 10 |
| Linear algebra and eigenvalues | 9 to 10 | Kreyszig, Ch 7 to 8 |
| Complex numbers | 11 | Kreyszig, Ch 13 |
| Differential equations and Laplace | 12 to 14 | Kreyszig, Ch 1 to 6 |
| Fourier analysis | 15 | Kreyszig, Ch 11 |
| Partial differential equations | 16 | Kreyszig, Ch 12 |
| Numerical methods | 17 | Kreyszig, Ch 19 to 21, with Chapra and Canale, Numerical Methods for Engineers |
| Probability and statistics | 18 | Kreyszig, Ch 24 to 25, with Montgomery and Runger, Applied Statistics and Probability for Engineers |
| Optimization | 19 | Kreyszig, Ch 22, with any engineering optimisation or design text |
Books to add for the foundations
Kreyszig is the spine, but it begins at differential equations and assumes the rest. These complete the course:
- Precalculus and trigonometry, for chapters 1 to 3: OpenStax Precalculus (free), or Stewart, Redlin and Watson, Precalculus.
- Calculus, for chapters 4 to 6: OpenStax Calculus, volumes 1 to 3 (free), or Stewart, Calculus: Early Transcendentals.
- Deeper linear algebra, optional: Strang, Introduction to Linear Algebra.
- Deeper numerical methods, optional: Chapra and Canale, Numerical Methods for Engineers.
- Deeper probability and statistics, optional: Montgomery and Runger, Applied Statistics and Probability for Engineers.
Optional extensions, for later
- Optimization and graphs, Kreyszig Ch 22 to 23: advanced and optional.