Advanced analysis, convergence and multivariable reasoning
Advanced analysis requires control of hypotheses and quantifiers, not only manipulation of formulas. Pointwise convergence allows the required sequence index to depend on the point. Uniform convergence controls the error simultaneously over the whole domain through a supremum bound. A moving narrow peak can tend to zero at every fixed point while its maximum remains unchanged. Before exchanging a limit with an integral or derivative, state the theorem that applies and verify its assumptions. Uniform convergence of continuous functions preserves continuity on the domain, but it does not by itself guarantee convergence of their derivatives.
For series of functions, a summable sequence of uniform absolute bounds gives the Weierstrass M-test. The domain matters: a bound valid away from a singular endpoint may fail on an interval containing it. On a finite closed interval, uniform convergence of continuous partial sums allows integration term by term. Dominated convergence uses a different sufficient condition: almost-everywhere convergence and one integrable dominating function. The dominating bound must work for the entire sequence and cannot be selected separately after taking each limit. Distinguish these sufficient criteria from necessary conditions and from a merely suggestive sketch.
Power series have a radius of absolute convergence, with boundary points requiring separate analysis. Ratio and root tests can identify the radius but an inconclusive test at a boundary does not determine divergence. Fourier series introduce a further distinction between a function and its periodic representation. Under suitable piecewise regularity, convergence at a jump is to the average of the one-sided limits. State the normalization of a Fourier transform before applying scaling or inversion, because factors involving2πdepend on the convention. Parseval identities likewise require a consistent coefficient convention and an appropriate function space.
In several variables, a stationary point has vanishing gradient but need not be an extremum. A positive-definite Hessian gives a strict local minimum and a negative-definite Hessian gives a strict local maximum under the usual twice-differentiable hypotheses. An indefinite Hessian gives a saddle. A semidefinite Hessian can leave the classification unresolved, but a genuinely positive quadratic direction already rules out a local maximum. Check higher-order behavior along null directions rather than treating every inconclusive test as permitting every classification.
Constrained optimization adds feasibility and constraint qualifications. Lagrange multiplier equations are candidate conditions, not a proof that every solution is a global optimum. Compare relevant candidates, boundaries and existence conditions. The implicit-function theorem requires a nonzero derivative in the variable solved for, or an invertible corresponding Jacobian in the vector case. Integral transformations require correct Jacobians and domains. Green, Stokes and divergence theorems connect local derivatives with boundary integrals under orientation and regularity assumptions. A zero curl implies a potential under suitable domain conditions; do not erase topology. These topics support actual advanced mathematical entrance practice, while extended proof construction remains a separate skill.