Multivariate Real Analysis

Rigorous analysis in ℝ^n: the total derivative as a linear map, the inverse and implicit function theorems, rigorous multivariable integration, and differential forms up to the general Stokes theorem.

Differentiation in Rⁿ

• Normed & Metric Spaces — Distance, generalized beyond the real line: the axioms and several norms.
• Total Derivative — The derivative as the linear map that best approximates a function.
• Jacobian Matrix — Pack all the partials into one matrix; its determinant is the local stretch.
• Chain Rule as Composition — The derivative of a composition is the matrix product of the derivative maps.
• Clairaut's Theorem — When mixed partials are continuous, the order of differentiation does not matter.

Inverse & Implicit Function Theorems

• Contraction Mapping Theorem — Banach's theorem: a contraction reels every point to one fixed point.
• Inverse Function Theorem — A nonzero Jacobian determinant certifies a local inverse, and the inverse's derivative is the matrix inverse.
• Implicit Function Theorem — When F(x,y)=0 defines y as a function of x, and what its slope is.
• Multivariable Taylor & Hessian Test — Second-order Taylor, the Hessian, and the discriminant test.

Integration & Forms

• Measure Zero & Lebesgue Criterion — Negligibly small made precise, and exactly which functions integrate.
• Fubini's Theorem — Compute a double integral one variable at a time, in either order.
• Partitions of Unity — Smooth weights that sum to one, gluing local pieces into a global whole.
• Differential Forms — The wedge product, the exterior derivative, and d^2 = 0.
• Stokes' Theorem on Manifolds — One theorem: integrate the derivative over the inside, integrate the form over the boundary.

Measure & Integration

• σ-Algebras — The family of sets you are allowed to measure.
• Measures — Assigning a consistent size to sets, and everything the two axioms force.
• Measurable Functions — Preimages of Borel sets, and a class that survives pointwise limits.
• The Lebesgue Integral — Slice the range, not the domain, and integrate what Riemann cannot.
• Convergence Theorems — When may a limit move inside an integral? Monotone convergence, Fatou, and domination.
• L^p Spaces — Norms on functions, and why membership depends on the exponent.
• Product Measures — Build a measure on a product, then swap the order of integration — and know when you may not.
• Radon-Nikodym — When one measure is another one reweighted, and the density that does the reweighting.

Not sure where to start? Take the ten-question placement test.