Ingénieur ITPE - independent mathematics, physics and infrastructure foundations cheat sheet
Ingénieur ITPE - independent mathematics, physics and infrastructure foundations · Keep the key ideas within reach.
Advanced analysis, convergence and multivariable reasoning
Control quantifiers and the domain in pointwise versus uniform convergence.
Verify hypotheses before exchanging limits, integrals or derivatives.
A positive Hessian direction rules out a local maximum.
Theorem normalizations, constraints and topology cannot be omitted.
Inner products, spectral reasoning and matrix structure
Distinguish similarity of maps from congruence of forms.
Projection coefficients follow residual orthogonality and the actual Gram matrix.
Repeated eigenvalues do not automatically imply diagonalizability.
Norm-spectral-radius equality is not an if-and-only-if test for normality.
Advanced probability, moments and conditional models
Include covariance in the variance of a linear combination.
Independence is stronger than zero covariance.
Bounds are not exact probabilities.
Limit, testing and interval conclusions require their stated assumptions.
Analytical mechanics, oscillation and model reduction
Define coordinates, frames, constraints and the system boundary.
Small-oscillation frequencies depend on mass as well as stiffness.
Momentum and kinetic-energy conservation have different conditions.
Fluid and rotating-frame formulas require their actual hypotheses.
Electromagnetic fields, thermal physics and wave models
Use geometry and boundary conditions, not symmetry by assertion.
An ideal magnetic force changes direction without doing work.
Reversible adiabatic and arbitrary adiabatic processes are different.
Thermal efficiency limits use absolute temperature and stated reservoir assumptions.
Geotechnical engineering models and applications
Effective stress equals total stress minus pore pressure
Use buoyant unit weight below the water table
Drained versus undrained strength depends on loading rate and permeability
USCS classification depends on grain size and plasticity
Active, at-rest, and passive earth pressures are distinct conditions
Water Resources and Environmental engineering models and applications
Use the correct rational-method unit conversion and an appropriate stated rainfall-intensity duration.
Manning's equation uses hydraulic radius A/P, not flow depth
Partial-full circular pipe flow can exceed full-flow velocity
Removal efficiency compares influent and effluent concentrations
Time of concentration sets the rainfall intensity duration
Construction engineering models and applications
Total float preserves the stated project finish; free float preserves successor early starts.
Bank, loose, and compacted volumes use swell and shrinkage separately
Critical path can shift after logic or duration changes
Crane capacity must be checked at the actual operating radius
Unit-price contracts pay by measured quantity; lump-sum pays for the scope
Ecology and Environmental Science
Logistic growth includes carrying capacity K; exponential growth does not.
Energy flows through ecosystems; 10 percent trophic transfer is an illustrative approximation, not a universal efficiency.
Most organisms require fixed nitrogen; phosphorus lacks a major atmospheric reservoir in the standard school-cycle model.
Mark–recapture: N = (marked first × total second) ÷ marked second.
Primary succession starts on bare rock; secondary succession starts where soil remains.