Concours ENSPY - independent subject practice cheat sheet
Concours ENSPY - independent subject practice · 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.