1. Engineering Mathematics 📐

Differential Equations

1st Order Linear: dy/dx + P(x)y = Q(x)
Integrating factor: μ = e^∫P(x)dx
Separable: dy/dx = f(x)g(y)

Laplace Transforms

L{f(t)} = ∫₀^∞ e^(-st)f(t)dt
L{1} = 1/s
L{e^at} = 1/(s-a)
L{sin(at)} = a/(s²+a²)

Numerical Methods

Newton-Raphson:

xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)

Euler's Method:

yₙ₊₁ = yₙ + h·f(xₙ,yₙ)

Simpson's Rule:

∫ ≈ (h/3)[f₀ + 4f₁ + f₂]

2. Fluid Mechanics 🌊

Concept	Equation	Application
Reynolds Number	Re = ρvD/μ = vD/ν	Flow regime (laminar/turbulent)
Bernoulli's	P/ρ + v²/2 + gz = const	Energy conservation in flow
Darcy-Weisbach	hf = f(L/D)(v²/2g)	Friction head loss in pipes
Hagen-Poiseuille	ΔP = 128μLQ/(πD⁴)	Laminar flow in pipes

Flow Regime:

Re < 2,100: Laminar | 2,100 < Re < 4,000: Transition | Re > 4,000: Turbulent

3. Heat Transfer 🔥

Conduction (Fourier's Law)

q = -kA(dT/dx)

For flat wall: Q = kA(T₁-T₂)/L

k = thermal conductivity (W/m·K)

Convection (Newton's Cooling)

q = hA(Ts - T∞)

h = convection heat transfer coefficient (W/m²·K)

Natural: 5-25 W/m²·K | Forced liquid: 100-20,000 W/m²·K

Radiation (Stefan-Boltzmann)

q = εσA(T₁⁴ - T₂⁴)

σ = 5.67 × 10⁻⁸ W/m²·K⁴ | ε = emissivity (0-1)

4. Mass Transfer 📦

Fick's First Law

JA = -DAB(dCA/dx)

Steady-state diffusion. DAB = diffusivity (m²/s)

Fick's Second Law

∂CA/∂t = DAB(∂²CA/∂x²)

Unsteady-state (transient) diffusion

5. Engineering Mechanics ⚙️

Key Concepts

Statics:

ΣF = 0 (force equilibrium)
ΣM = 0 (moment equilibrium)
Moment = Force × Distance

Stress-Strain:

σ = F/A (stress)
ε = ΔL/L (strain)
E = σ/ε (Young's modulus)

6. Practice Questions 📚

Common Board Exam Questions

Q1: Water flows in a 5cm pipe at 2 m/s. If viscosity = 0.001 Pa·s and density = 1000 kg/m³, what is Re?

A: Re = ρvD/μ = (1000)(2)(0.05)/0.001 = 100,000 (turbulent)

Q2: Heat transfers through a 10cm wall (k=1.5 W/m·K) with T₁=100°C, T₂=20°C. Find heat flux.

A: q/A = k(ΔT)/L = 1.5(100-20)/0.1 = 1200 W/m²

Q3: Apply Newton-Raphson to find √2 starting from x₀=1.

A: f(x)=x²-2, f'(x)=2x. x₁ = 1 - (1-2)/(2) = 1.5. x₂ = 1.5 - (2.25-2)/3 = 1.417

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General Engineering