About This Course

This book extends calculus from functions of one variable to functions of several variables and to vector fields in the plane and in space. It is written for a reader working alone: a student who has finished a first course in calculus and a first course in linear algebra and who wants to compute confidently, to know why each formula holds, and to know exactly when it stops holding.

Three things are assumed. From Calculus: limits, derivatives, the chain rule, the definite integral, substitution, integration by parts, and the Fundamental Theorem of Calculus. From Linear Algebra: vectors, the dot product, matrices as linear maps, matrix multiplication, determinants, and eigenvalues of symmetric matrices. From a first encounter with proof: comfort with quantifiers and with the phrase “for every small enough”. A handful of results about continuity and compactness are owned by Real Analysis. When one of them is needed it is stated with the hypotheses the argument uses and marked once with its library identifier; it is not proved again here.

Six skills are the measure of having finished, each performed without looking anything up. Describe curves, planes, and surfaces with vectors and compute velocity, arc length, and curvature. Compute partial derivatives, gradients, and Jacobian matrices, and explain the derivative of a map as the linear map that best approximates it. Find and classify critical points, with and without constraints, and say what guarantees that an extremum exists. Set up and evaluate double and triple integrals in whichever order and coordinate system suits the region. Compute work and flux integrals from a parametrization. Use the theorems of Green, Stokes, and Gauss to trade an integral over a region for one over its boundary, after checking smoothness and orientation.

Notation is conventional. Vectors and points of \(\mathbb R^n\) are bold, \(\mathbf a=(a_1,\dots,a_n)\) , and are treated as columns whenever a matrix acts on them. The dot product is \(\mathbf a\cdot\mathbf b\) , the Euclidean norm is \(\lVert\mathbf a\rVert=\sqrt{\mathbf a\cdot\mathbf a}\) , and the standard basis of \(\mathbb R^3\) is \(\mathbf i,\mathbf j,\mathbf k\) . Partial derivatives are written \(\partial f/\partial x\) or \(f_x\) . A function is of class \(C^1\) on an open set if its first partial derivatives exist and are continuous there, and of class \(C^2\) if the same holds for its second partial derivatives.

Each chapter ends with exercises that run from recall through computation to proof, and each set contains at least one argument with a flaw to be found. Work them with pencil before looking anything up. The companion solutions give complete arguments for some exercises, graduated hints for others, and bare answers for a few. The cumulative review at the end mixes the chapters without saying which tool each problem needs, because choosing the tool is part of the problem.