Concours ENSPY - independent subject practice common mistakes
Concours ENSPY - independent subject practice · Understand the traps and how to avoid them.
Interchanging a limit and integral from pointwise convergence alone
Pointwise convergence does not supply the uniform or integrable control needed by the relevant theorem.
What to do instead
Name a sufficient convergence theorem and verify its assumptions on the exact domain before exchanging the operations.
Treating a semidefinite Hessian as permitting every extremum
A positive quadratic direction can already rule out a maximum even when a zero direction leaves a minimum or saddle unresolved.
What to do instead
Check known directions, then analyze higher-order behavior in the null directions without contradicting the nonzero curvature.
Using a sufficient spectral condition as a characterization
Normality gives Euclidean-norm/spectral-radius equality, but the equality can hold for some non-normal matrices.
What to do instead
State the implication direction and test a block-matrix counterexample before asserting an if-and-only-if theorem.
Computing a projection with the wrong inner product
Euclidean coordinate intuition can differ from a weighted or integral inner product.
What to do instead
Build the correct Gram system and verify that the residual is orthogonal to every basis vector.
Assuming Newton converges from every starting point
A small derivative, multiple root or unsuitable initialization can invalidate an expected local-convergence argument.
What to do instead
Check the hypotheses, use a bracket or safeguarded method where appropriate and assess a meaningful residual and error criterion.
Equating a stable numerical step with an accurate solution
Stability controls growth but does not itself make truncation error acceptable.
What to do instead
Check the stability restriction, compare mesh refinements and evaluate accuracy against the specified tolerance.
Applying conservation outside its system boundary
External forces, torque, heat or work can change the conserved quantity for the selected system.
What to do instead
Define the system and frame, identify exchanges and state the assumptions supporting each conservation relation.
Assuming adiabatic always means constant entropy
Irreversible processes can generate entropy even when heat transfer is absent.
What to do instead
Distinguish no heat transfer from reversibility and track entropy generation under the actual model.