Calculus & Differential Equations
Maxima/minima, vector calculus theorems and PDE classification — the toolbox questions shared by every branch.
Key formulas
D = fxx·fyy − fxy² at the stationary point
Second-derivative test (2 variables): Classify critical points without plotting.
dy/dx + P(x)·y = Q(x) → IF = e^(∫P dx)
Integrating factor: Multiply through; the left side becomes d/dx(y·IF).
∮_S F·n dS = ∭_V (∇·F) dV
Divergence theorem: Total outflow equals the volume integral of the divergence — used in fluid/flux questions.
∮_C F·dr = ∬_S (∇×F)·n dS
Stokes / Green: Circulation equals the surface integral of curl (Stokes; Green is its 2D case).
Cheat sheet
- ▸Maxima/minima: solve fx = fy = 0, then D = fxx·fyy − (fxy)²: D > 0, fxx > 0 minimum; D > 0, fxx < 0 maximum; D < 0 saddle.
- ▸Gradient ∇f points along maximum increase and is normal to level curves/surfaces. Directional derivative = ∇f · û.
- ▸Green's theorem (2D plane loop) → Stokes (curl through a surface) → Gauss/divergence (flux through closed surface). Memorise what each converts: loop↔area, surface↔loop, closed surface↔volume.
- ▸First-order ODEs: separable, linear (integrating factor e^∫P dx), exact (M dx + N dy with ∂M/∂y = ∂N/∂x), homogeneous (substitute y = vx).
- ▸Second-order linear with constant coefficients: solve characteristic equation; real distinct → two exponentials, repeated → multiply one by x, complex → e^(αx)(C cos βx + D sin βx).
- ▸PDE classification: B² − AC < 0 elliptic (Laplace — steady state), = 0 parabolic (heat/diffusion), > 0 hyperbolic (wave).
PDE classification
| Type | Condition (B² − AC) | Example | Behaviour |
|---|---|---|---|
| Elliptic | < 0 | Laplace ∇²u = 0 | Steady-state, smooth solutions |
| Parabolic | = 0 | Heat u_t = αu_xx | Diffusion, smooths over time |
| Hyperbolic | > 0 | Wave u_tt = c²u_xx | Propagates waves, sharp fronts |
Visual
Critical point test:
fyy>0 & D>0 D<0
(bowl / min) (saddle)
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Gradient ∇f ⟂ level curves, points uphill (steepest ascent).Top 4 campus interview questions
MCQ mastery quiz
1. At a stationary point, D = fxx·fyy − (fxy)² < 0 indicates…
2. The gradient of f at a point is always…
3. The PDE u_tt = c²u_xx is classified as…
4. The integrating factor of dy/dx + 2y = e^x is…