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Engineering Mathematics

From transcript: 20MA110 Eng Math 1 (D) · 20MA210 Eng Math 2 (E) · 20MA311 Eng Math 3 (C) · 20MA411 Math (A)

Cheat sheet

Printable study sheet: formulas → definitions → topic notes → traps.

Formulas

ODE 1st linear: dy/dx + P(x)y = Q(x) ; IF = e^{∫P dx}
2nd order const coeff: y''+ay'+by=0 → r²+ar+b=0 ; over/crit/under damp if disc >/=/<0
Laplace: L{f'}=sF−f(0); L{f''}=s²F−sf(0)−f'(0); L{e^{at}f}=F(s−a)
Fourier series (period 2π): a0/2 + Σ(an cos nx + bn sin nx)
grad φ = ∂φ/∂x i + ∂φ/∂y j + ∂φ/∂z k ; div F = ∇·F ; curl F = ∇×F
Green: ∮P dx+Q dy = ∬(∂Q/∂x−∂P/∂y) dA ; Stokes: ∮F·dr = ∬(∇×F)·dS ; Divergence: ∯F·dS=∭∇·F dV
Line integral work W=∫F·dr ; conservative iff curl F=0 (simply connected)
Eigen: Ax=λx ; det(A−λI)=0 ; modal FEA: Kφ=ω²Mφ
PDE heat: ∂u/∂t = α ∇²u ; wave: ∂²u/∂t² = c² ∇²u ; Laplace: ∇²u=0
Separation of variables + BCs → eigenvalues / Fourier coefficients
Taylor: f(a+h)=f(a)+hf'+h²f''/2!+… ; used in linearization & numerical methods
Complex: e^{iθ}=cosθ+i sinθ ; useful for AC phasors & harmonic solutions
Probability: E[X]=Σ x p(x) ; Var=E[X²]−(E[X])² ; normal Z=(x−μ)/σ
Least squares: minimize ||Ax−b||² → AᵀA x̂ = Aᵀb (normal equations)
Numerical: Newton x_{n+1}=x_n−f(x_n)/f'(x_n) ; trapezoidal / Simpson integration
∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z (divergence)
∇×F = |i j k; ∂/∂x ∂/∂y ∂/∂z; P Q R| (curl)
Laplace: L{f'(t)} = s F(s) − f(0)
Fourier: f(x) = a0/2 + Σ (an cos nx + bn sin nx)
ODE: ay'' + by' + cy = 0 → r = (−b ± √(b²−4ac))/(2a)

Definitions

Order of ODE
Highest derivative present; need that many ICs/BCs
Linear ODE
y and derivatives appear to first power only — superposition holds
Eigenvalue / eigenvector
Direction unchanged by A; λ = stretch; vibration ω² ↔ λ
Orthogonal functions
∫ φ_m φ_n = 0 (m≠n) — basis for Fourier / Sturm–Liouville
Gradient
Direction of steepest ascent of scalar field
Divergence
Net outward flux per volume — source strength
Curl
Local rotation / circulation density
Laplace transform
Maps DE + ICs to algebra in s-domain (controls, circuits)
Ill-conditioned matrix
Small input change → large solution change; check units/mesh in FEA
PDE type
Elliptic (steady Laplace), parabolic (diffusion), hyperbolic (wave)
Gradient ∇f
Vector of steepest ascent of scalar field f
Eigenvalue
Av = λv — stretch factor along eigenvector
PDE types
Elliptic (Laplace), parabolic (heat), hyperbolic (wave)

Topic-wise short notes

Study these first — one block per syllabus topic. Then read the deep notes below.

Calculus & differential equations

  • Model: free vibration mẍ+cẋ+kx=F(t); characteristic r from assuming e^{rt}.
  • Homogeneous solution + particular solution; resonance when forcing ≈ natural freq.
  • Laplace turns derivatives into multiplication by s — standard for control TFs.
  • First-order linear: integrating factor method is the workhorse.
  • Nonlinear ODEs rarely closed-form — linearize about operating point for controls.

Vector calculus

  • Continuity (mass): ∂ρ/∂t + ∇·(ρV)=0 ; incompressible ∇·V=0.
  • Strain ~ symmetric part of ∇u ; rotation ~ skew part.
  • Stokes links line circulation to surface curl — lift/circulation ideas.
  • Divergence theorem converts surface flux integrals to volume — CFD residuals.
  • Practice identities: ∇×(∇φ)=0 ; ∇·(∇×A)=0.

Laplace & Fourier

  • Unit step, ramp, impulse, exponential, sin/cos transforms — memorize table.
  • Final value theorem: lim t→∞ f = lim s→0 sF(s) (if poles left-half).
  • Fourier series for periodic BCs on rods/plates; FFT for signal/vibration spectra.
  • Half-range expansions for odd/even extensions on [0,L].

ODEs & PDEs for engineers

  • Heat equation: temperature diffusion; Fourier number Fo=αt/L².
  • Wave equation: strings, acoustics, 1-D stress waves c=√(E/ρ).
  • Laplace equation: steady temperature / potential flow.
  • Separation: product solutions X(x)T(t); eigenvalues from homogeneous BCs.

Linear algebra in FEA

  • Global K assembled from element matrices; sparse symmetric positive definite (with BCs).
  • Modal analysis solves generalized eigenproblem; mode shapes orthogonal w.r.t. M,K.
  • Conditioning: mixed units, very stiff springs, poor aspect ratios hurt solvers.
  • Interview: residual r=b−Ax measures equilibrium error.

Statistics for engineers

  • Normal, binomial; confidence intervals on mean.
  • Linear regression & R²; residual plots for model quality.
  • Hypothesis testing idea: fail to reject vs reject H0.

Exam traps & quick notes

Comprehensive notes

Calculus & differential equations

Chain rule, Taylor expansions, and 1st/2nd-order linear ODEs underpin beam vibration, RC circuits analog, and transient heat. Homogeneous + particular solution; undetermined coefficients / variation of parameters.

External: MIT OCW Diff Eq ↗ Khan Academy Calculus ↗

Vector calculus

Line/surface integrals, Green/Stokes/Divergence theorems connect flux and circulation — language of continuum mechanics and Maxwell/fluid conservation laws.

External: 3Blue1Brown Essence of Linear Algebra ↗ Paul's Online Notes Calc III ↗

Laplace & Fourier

Laplace for IVPs and transfer functions G(s)=Y/U. Fourier for periodic signals and PDE separation of variables.

External: Control tutorials (CTMS) ↗

ODEs & PDEs for engineers

First/second-order linear ODEs model free vibration (mẍ+cẋ+kx=F). Characteristic equation, over/under/critical damping. Laplace transforms turn IVPs into algebra — foundation for control transfer functions. PDEs: heat equation, wave equation, Laplace equation; separation of variables + Fourier series appear in conduction and acoustics.

External: NPTEL ↗ Engineering Toolbox ↗

Linear algebra in FEA

Ax=b is the discrete FEA problem. Eigenvalues = natural frequencies in modal analysis (Kφ=λMφ). Ill-conditioning from poor mesh/units. Interview: explain residual and why boundary conditions reduce DOF.

External: NPTEL ↗ Engineering Toolbox ↗

Interview Q&A for this subject

Q: What is an eigenvalue in physical terms?
A: Stretch factor of a linear transform; in vibration, related to ω² for mode shapes.

External: ASQ / quality ↗ MIT OCW ↗