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
- algebraic operations with vectors, basic properties
- inner products, 1-norm, -norm, -norm, distances, etc.
- Cauchy-Schwarz inequality, triangle inequalities

- sequences in
- epsilon-delta definitions of convergence, Cauchyness and boundedness
- component sequences, Cauchy criterion, Bolzano-Weierstrass theorem

- topology in
- 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 ; closure is the smallest closed set containing )
- 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 with
- epsilon-delta definition of continuity,
*equivalence*to “respecting convergent sequences” (contradiction proof) - the component functions of , denoted
- component functions are continuous
*if and only if*is continuous, etc. - linear combination of continuous functions is again continuous (verified using sequences)

- epsilon-delta definition of continuity,
- relationships between continuity and compactness
- epsilon-delta definition of uniform continuity (stronger notion than continuity – choice of does not depend on the point 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 (simple corollary of the above)

- integrability of bounded functions with
- closed cells in , their volume, diameter, and -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 with respect to a division
- Darboux sum inequalities involving refinements
- for any two divisions (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 with as
- 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 , used to define the volume of

- 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

- functions
- weak MVT lemma, “big quotient lemma”
- linear combination theorem for directional derivatives (relies on two previous lemmas)
- strong MVT for

- chain rule for where 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 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)
- is an
*algebra of functions*+ directional derivative theorems for algebraic operations with functions

- functions in
- 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 manifolds of dimension

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