## 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 $\mathbb{R}^n$
• algebraic operations with vectors, basic properties
• inner products, 1-norm, $p$-norm, $\infty$-norm, distances, etc.
• Cauchy-Schwarz inequality, triangle inequalities
• 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 $d$