Concours ATS - independent subject practice cheat sheet
Concours ATS - 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.
Statics, strength and mechanical transmission
In an ideal planar model: pin has two force components, roller one normal reaction, fixed support two force components and a moment.
I = b·h³/12, W = b·h²/6 for a rectangle
Simply supported beam, central load: M_max = F·L/4
Euler: P_cr = π²EI/L_k², with L_k depending on the end conditions
Planar DOF: G = 3(n−1) − 2j₁ − j₂
For the defined ideal lossless gear pair, speed and torque-magnitude ratios are inverse; losses reduce actual output torque.
Measurement, feedback, PLC and automation
Sensitivity = Δoutput/Δinput; resolution = smallest detectable change
Closed-loop error compares reference and measurement; integral action can remove appropriate steady-state error only under suitable stability and operating assumptions.
PLC scan: read inputs → run program → update outputs
4–20 mA live zero: (I − 4)/16 of the span
Repeatability ≠ accuracy, resolution ≠ sensitivity
Robot DOF come from its joints; workspace is the reachable volume
Building systems, load paths, heat and moisture
R=d/λ uses thickness in metres; total resistance includes the model’s surfaces and layers.
U=1/R_total is a steady-state element model, not the complete building energy demand.
Permanent, occupancy, wind and snow actions require the stated design framework.
Moisture, airtightness and intended ventilation must be considered together.
Example tie spacing is supplied data, not a code rule.
Technical English, scientific claims and engineering communication
Read modals, conditions, negation and logical connectors carefully.
Keep units, denominators and percentage-point distinctions.
Paraphrases must preserve qualification and scope.
A written technical exercise does not certify practical or oral competence.