Multivariable Calculus
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This resource develops multivariable calculus — partial derivatives, gradients, Jacobians, line and surface integrals, and the theorems of Green, Stokes, and Gauss — carefully, with full proofs and the geometry made explicit throughout.
The approach differs from most treatments in one respect: rather than introducing coordinates immediately and defining partial derivatives as the primary objects, we begin with the total derivative — the linear map best approximating a smooth function near a point — and treat the Jacobian matrix as its coordinate representation. Partial derivatives, gradient, divergence, and curl then emerge as special cases, distinguished by the dimension and structure of domain and codomain.
This order of presentation has a cost and a benefit. The cost: the first few chapters are more abstract than a standard multivariable course. The benefit: the classical objects fit into a coherent picture, and results like the chain rule, change of variables, and the integral theorems become consequences of a single idea rather than separate facts to memorize. Students who have found multivariable calculus a disconnected list of techniques may find this framing clarifying.
To make the derivative-as-linear-map idea precise, we need a minimal amount of topology and the notion of a smooth manifold. These are introduced in the first two chapters — enough to state things correctly, not enough to become a course in differential geometry. We work primarily with manifolds embedded in \mathbb{R}^n, curves and surfaces in \mathbb{R}^3 being the central cases.
Contents
Topology of \mathbb{R}^n: Open and closed sets, limits, continuity, compactness — the analytic foundations underlying all subsequent results.
Smooth Manifolds: Curves, surfaces, and higher-dimensional smooth spaces embedded in \mathbb{R}^n; tangent spaces; smooth maps.
Differentiation: The total derivative as a linear map, the Jacobian as its coordinate representation, partial derivatives, gradient, divergence, curl, and the classical theorems: Clairaut, inverse function, implicit function.
Integration: Iterated integrals, Fubini’s theorem, change of variables, line integrals, surface integrals.
The Integral Theorems: Green’s theorem, the Divergence theorem, Stokes’ theorem — understood as instances of a single idea relating integrals over a region to integrals over its boundary.
Outlook: A closing chapter sketches how the theory extends beyond \mathbb{R}^3 and points toward differential forms and abstract manifolds.
How to Read This
The chapters are designed to be read in order. That said:
- For standard multivariable calculus: the topology and manifold chapters (Ch. 1–2) can be read lightly on a first pass, returning when precision is needed. The differentiation material (Ch. 3 onward) is self-contained for readers willing to accept the setup informally.
- For a more geometric treatment: read Ch. 1–2 carefully. The payoff is a unified view of differentiation and integration that makes the integral theorems transparent.
Prerequisites
Single-variable calculus and linear algebra at the level of the preceding texts in this series. The treatment draws on vector spaces and linear maps throughout — in particular, the Jacobian is treated as a matrix representing a linear map, a distinction developed carefully in the Linear Algebra text.