247 Review Post #1

So I know I should have gotten around to posting a follow-up to my (not-so-)recent post on the theory of single variable calculus a long time ago, but this (and the next few posts) will be regarding MATH 247 (multivariable calculus) since that’s my first exam.

  • structure on \mathbb{R}^n
  • sequences in \mathbb{R}^n
    • epsilon-delta definitions of convergence, Cauchyness and boundedness
    • component sequences, Cauchy criterion, Bolzano-Weierstrass theorem
  • topology in \mathbb{R}^n
    • interior points, open sets, closed sets, the “cannot escape” property
    • interior, closure, and boundary of a set, relationships between complements (interior is the largest open set contained within A; closure is the smallest closed set containing A)
    • a set is closed if and only if it has the “cannot escape” property (contradiction proof)
    • compactness and its implications with respect to the convergence of sequences (Bolzano-Weierstrass + “cannot escape” property)
  • continuity of functions f : A \to \mathbb{R}^m with \emptyset \neq A \subseteq \mathbb{R}^n
    • epsilon-delta definition of continuity, equivalence to “respecting convergent sequences” (contradiction proof)
    • the m component functions of f, denoted f^{(i)}
    • component functions are continuous if and only if f is continuous, etc.
    • linear combination of continuous functions is again continuous (verified using sequences)
  • relationships between continuity and compactness
    • epsilon-delta definition of uniform continuity (stronger notion than continuity – choice of \delta does not depend on the point \vec{a} \in A in question)
    • in the special case of a compact set, continuity implies uniform continuity (contradiction proof via Bolzano-Weierstrass)
    • the continuous image of a compact set is compact (also a contradiction proof via Bolzano-Weierstrass)
    • Extreme Value Theorem for functions f : A \to \mathbb{R}^m (simple corollary of the above)
  • integrability of bounded functions f : A \to \mathbb{R} with A \subseteq \mathbb{R}^n
    • closed cells in \mathbb{R}^n, their volume, diameter, and \infty-diameter
    • division of a closed cell, mesh and refinements, common refinements
    • supremum and infimum of a bounded function over a subset of its domain
    • upper and lower Darboux sums of f with respect to a division \Delta
    • Darboux sum inequalities involving refinements
    • L(f, \Delta_1) \leq U(f, \Delta_2) for any two divisions \Delta_1, \Delta_2 (simple application of the above)
    • upper and lower integrals defined as infimums and supremums of upper and lower Darboux sums, respectively
    • definition of integrability and the equivalent epsilon-Delta formulation
    • for integrable functions, there exist a sequence of divisions \Delta_k with U(f,\Delta_k) - L(f,\Delta_k) \to 0 as k \to \infty
    • sums and constant multiples of integrable functions are integrable, inequalities relating supremums and infimums
    • the vector space of integrable functions on a closed cell
  • continuity and integrability
    • null sets
    • bounded functions that are continuous everywhere on a closed cell except for a null set are integrable (proof)
    • the characteristic function of A, used to define the volume of A
  • Fubini’s theorem (for multiple integrals)
    • decomposition of closed cells
    • partial functions
    • Fubini’s theorem: allows reduction of multiple integrals to iterated integrals under certain conditions
    • establishing a grid from two divisions by “extending” the boundaries
    • lemmas involving inequalities and grids
    • use of these lemmas to prove Fubini’s theorem
    • examples where one or both of the hypotheses of Fubini’s theorem do not hold
  • integration on more general domains
    • if a function is defined on an arbitrary set, we “cover” that set with a cell, and define its value to be 0 outside of its domain – then the function is integrable on this cell if and only if it is integrable on its domain
    • furthermore, the “covering cell” chosen does not affect the integral
    • a continuous function defined on a set of null boundary is integrable
    • examples of integration on “weird” domains
    • if the constant function 1 is integrable on an arbitrary set, we say the set has volume, and define the volume of that set to be this integral
    • change of variables (?)
  • partial derivatives
    • homogeneity of directional derivative
  • \mathcal{C}^1 functions
    • weak MVT lemma, “big quotient lemma”
    • linear combination theorem for directional derivatives (relies on two previous lemmas)
    • strong MVT for \mathcal{C}^1(A, \mathbb{R})
  • chain rule for (f \circ \varphi) where \varphi is a differentiable path (based on the existence of a Lipschitz constant on a convex, compact domain)
  • higher-order partial derivatives
    • Clairaut’s theorem for commutativity of mixed partial derivatives (based on \alpha, \beta existence lemma)
    • Hessian matrix test for local minima/maxima (multivariable analogue of second derivative test)
    • tangent hyperplanes to a graph and the normal vector (calculated via the gradient)
    • \mathcal{C}^1(A,\mathbb{R}) is an algebra of functions + directional derivative theorems for algebraic operations with functions
  • functions in \mathcal{C}^1(A, \mathbb{R}^m)
    • chain rule for Jacobians (proved via chain rule for differentiable paths)
  • inverse and implicit function theorem
    • “stability under small perturbation” lemma for matrix invertibility
    • inverse function theorem
    • implicit function theorem and parametrized \mathcal{C}^1 manifolds of dimension d

About mlbaker

just another guy trying to make the diagrams commute.
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