Multivariable and Vector Calculus

Theory part 3Practice part 4

dot product, length and angle, orthogonality, direction cosines, scalar and vector projection, work and Cauchy-Schwarz inequality

Also uses Vector components, magnitude, unit vectors, elementary trigonometry and algebra.

Theory part 5Practice part 6

cross product in three dimensions, right-hand rule, perpendicularity, parallelogram area, torque, scalar triple product and signed and geometric volume

Also uses Vectors in an orthonormal right-handed coordinate system, dot product, sine and cosine.

Theory part 7Practice part 8

matrix notation, matrix-vector multiplication, composition, linear systems, invertible matrices, determinant, area and volume scaling, orientation and basis changes

Also uses Vector components, linear combinations, basic algebra and simultaneous equations.

Theory part 9Practice part 10

vector and parametric line equations, symmetric form with zero-component care, skew lines, plane normal, point-plane distance, angles and intersections

Also uses Vector arithmetic, dot and cross products, linear equations and elementary trigonometry.

Theory part 11Practice part 12

traces, cylinders, ellipsoids, elliptic and hyperbolic paraboloids, cones, one-sheet and two-sheet hyperboloids, translations and identifying surfaces

Also uses Coordinate planes, equations of conic sections, completing the square and square roots.

Theory part 13Practice part 14

cylindrical coordinates, spherical coordinates, coordinate conversion, azimuth and polar-angle convention, domains, coordinate surfaces and nonuniqueness at singular coordinates

Also uses Cartesian coordinates, polar coordinates, sine, cosine and basic surface equations.

Theory part 15Practice part 16

component functions, limits, continuity, space curve, helix, orientation, intersection curves and parameter domain

Also uses Functions, scalar limits and continuity, Cartesian coordinates, elementary trigonometric identities.

Theory part 17Practice part 18

componentwise differentiation and integration, tangent vector, regular curves, unit tangent, product rules for dot and cross products and vector antiderivative

Also uses Scalar derivatives and integrals, vector functions, dot and cross products, basic limits.

Theory part 19Practice part 20

speed, arc length function, arc length parameter, regularity, reparametrization, orientation and invariance of geometric length

Also uses Vector derivatives, definite integrals, scalar chain rule and inverse functions.

Theory part 21Practice part 22

curvature, osculating circle, principal unit normal, binormal, tangent-normal-binormal frame, normal and osculating planes, torsion and nonzero-speed and curvature conditions

Also uses Regular vector curves, arc length, vector derivatives, dot and cross products.

Theory part 23Practice part 24

velocity, speed, acceleration, tangential and normal acceleration, projectile motion, central forces, angular momentum and Kepler's laws with inverse-square-force assumptions

Also uses Vector derivatives, dot/cross products, tangential-normal frame, elementary integration and conic geometry.

Theory part 25Practice part 26

domain and range, scalar field, graphs, traces, level curves, contour maps, level surfaces, open and closed sets, boundary and boundedness

Also uses Coordinates in the plane and space and Real functions, square roots, and logarithms.

Theory part 27Practice part 28

epsilon-delta definition using distance, all approaches, path tests can disprove a limit, agreeing paths do not prove it, squeeze estimates and continuity relative to a domain

Also uses Functions of several variables, Euclidean distance and Single-variable limits.

Theory part 29Practice part 30

varying one coordinate, slopes of traces, partial derivative notation, higher partial derivatives, mixed partials, Clairaut's theorem under continuity hypotheses and partial differential equations

Also uses Single-variable differentiation and chain rule and Functions, domains, and traces.

Theory part 31Practice part 32

differentiability as a linear map plus o(norm of increment), Jacobian matrix, tangent plane, linearisation, total differential, partial derivatives alone are insufficient and sufficient continuous-partial condition

Also uses Partial derivatives, Linear maps and matrices and Limits using Euclidean distance.

Theory part 33Practice part 34

composition of multivariable functions, dependency diagrams, Jacobian multiplication, one and several independent parameters and total versus partial derivative

Also uses Differentiability and Jacobian matrices, Partial derivatives and One-variable chain and product rules.

Theory part 35Practice part 36

unit direction, directional limit, gradient formula for differentiable functions, steepest ascent, level-set normal, gradient zero and directional derivatives alone are insufficient

Also uses Partial derivatives and differentiability, Dot product and unit vectors and Chain rule.

Theory part 37Practice part 38

C1 regularity on a neighbourhood, implicit equations, local solvability, nonzero relevant partial or invertible Jacobian, local versus global inverse, derivatives of local branches and tangent planes to level surfaces

Also uses Partial derivatives and C1 regularity, Chain rule and Jacobian matrices and matrix inverses.

Theory part 39Practice part 40

second-order Taylor polynomial, Hessian, quadratic forms, higher-order multi-index notation, remainder, symmetry under regularity conditions and approximation error

Also uses Differentiability and gradients, Second partial derivatives and One-variable Taylor theorem.

Theory part 41Practice part 42

critical points, local maxima and minima, saddles, Hessian definiteness, two-variable second derivative test and degenerate and inconclusive cases

Also uses Gradient and partial derivatives, Second-order Taylor approximation and Quadratic forms.

Theory part 43Practice part 44

continuous function on a closed bounded region, interior critical points, boundary curves, corners, constrained domains and open and unbounded regions

Also uses Functions and domain geometry, Critical points and Single-variable optimization.

Theory part 45Practice part 46

constrained extrema, one or several equality constraints, parallel gradients, independent constraint gradients, singular constraint points, feasible boundaries and candidate verification

Also uses Gradients and regular level sets, Absolute extrema and Linear independence and dot products.

Theory part 47Practice part 48

double Riemann sums, signed volume, integrability, average value, iterated integrals and Fubini under continuity or absolute integrability

Also uses Single-variable definite integrals and antiderivatives and Functions of two variables and rectangular coordinates.

Theory part 49Practice part 50

type I and type II descriptions, splitting regions, bounds from geometry, changing order, bounded piecewise continuous integrands and improper cases

Also uses Double integrals over rectangles and Fubini under continuity and Solving boundary equations and one-variable substitution.

Theory part 51Practice part 52

polar area element r dr dtheta, sector area scaling, polar regions, coordinate range, bounds and avoiding overlap

Also uses Polar coordinate conversion with nonnegative radius and Double integrals and elementary trigonometric integration.

Theory part 53Practice part 54

area, density and mass, moments, centre of mass, moment of inertia, radius of gyration, joint probability density, marginals, expectations and covariance

Also uses Double integrals and polar coordinates and Weighted averages and basic probability language.

Theory part 55Practice part 56

triple Riemann sums, iterated integrals, region descriptions, changing order, volume, mass, moments and average value

Also uses Double integrals and iterated bounds and Three-dimensional coordinate geometry and density.

Theory part 57Practice part 58

volume element r dr dtheta dz, cylindrical regions, integration order, bounds and symmetry

Also uses Polar coordinates and triple integrals and Cylinders, cones and paraboloids.

Theory part 59Practice part 60

volume element rho squared sin(phi), polar angle from the positive z-axis, spherical regions, radial and angular bounds and symmetry

Also uses Spherical coordinate conversion and Triple integrals and elementary trigonometric substitution.

Theory part 61Practice part 62

coordinate transformation, image region, derivative matrix, absolute Jacobian determinant, local invertibility, one-to-one coverage except boundary exceptions and double and triple integrals

Also uses Derivative matrices and determinants, Double and triple integrals and Coordinate transformations.

Theory part 63Practice part 64

unbounded regions and singular integrands, exhaustion by regions, continuous nonnegative integrands, elementary improper-integral form of Tonelli, Fubini under absolute integrability, failures without hypotheses, Gaussian integral and distinction from measure-theoretic generalisations

Also uses Improper one-variable integrals and convergence tests and Double integrals, polar coordinates and change of variables.

Theory part 65Practice part 66

plane and space vector fields, velocity field, force field, gradient field, potential, field lines and domain and singularities

Also uses Vectors, derivatives of curves, gradients and elementary differential equations.

Theory part 67Practice part 68

integration with respect to arc length, piecewise regular curves, density on a wire, parametrization independence and orientation invariance of scalar length integrals

Also uses Arc length and regular parametrized curves and Scalar density and one-variable integration.

Theory part 69Practice part 70

vector line integral, differential dr, coordinate forms, work, circulation, reversing orientation and parametrization

Also uses Dot products, parametrized curves and arc length and Scalar line integrals and basic work.

Theory part 71Practice part 72

potential difference, path independence, closed-loop integrals, finding potentials, connected open domains and curl-free sufficient on simply connected domains with C1 regularity

Also uses Vector line integrals, gradients and the chain rule and Open domains and elementary mixed partial derivatives.

Theory part 73Practice part 74

divergence as local source strength, curl as local circulation, del operator, Laplacian, divergence of curl, curl of gradient and smoothness hypotheses

Also uses Partial derivatives, gradients and mixed-partial symmetry and Dot products, cross products and line-integral orientation.

Theory part 75Practice part 76

circulation and flux forms, positively oriented piecewise smooth boundary, C1 field on a neighbourhood of the region, simple regions and holes and area formula

Also uses Vector line integrals, divergence and planar curl and Double integrals, boundary orientation and regular planar regions.

Theory part 77Practice part 78

surface parametrization, grid curves, tangent plane, cross product of partial derivatives, regular surface, surface area of a graph and overlap and coverage

Also uses Partial derivatives and differentiability, Cross products and double integrals and Three-dimensional coordinate systems.

Theory part 79Practice part 80

surface integral, surface element, density, oriented surface, unit normal, vector flux, orientation reversal, nonorientable surfaces and closed-surface outward normal

Also uses Regular surface parametrizations and surface area and Dot products, cross products and multiple integrals.

Theory part 81Practice part 82

oriented piecewise smooth surface, induced boundary orientation, circulation and curl flux, C1 field on a neighbourhood and compatible normal and boundary direction

Also uses Vector line integrals, curl, oriented surface flux, Green's theorem and regular parameterized surfaces.

Theory part 83Practice part 84

outward flux, divergence over a solid, closed oriented boundary, regularity assumptions, cavities, excluded singularities, Gauss's law and relationship among integral theorems

Also uses Divergence, vector surface flux, multiple integrals, outward orientation and the fundamental theorem of calculus.

Theory part 85Practice part 86

differential form M dx+N dy, potential function, exactness, continuous partial derivatives, simply connected domain condition, integrating factors and implicit solutions

Also uses Partial derivatives and the chain rule, Antiderivatives and elementary first-order differential equations and Parametrized curves and line integrals.

Theory part 87Practice part 88

real and imaginary parts, complex plane, modulus and argument, conjugates, arithmetic, polar form, Euler's formula from power series, complex exponential and real sinusoidal solutions

Also uses Real algebra and trigonometry, Power series and ordinary derivatives and The meaning of a differential-equation solution.

Theory part 89Practice part 90

linear homogeneous equation, superposition, independent solutions, Wronskian, constant-coefficient characteristic equation, distinct and repeated real roots, conjugate complex roots, initial-value and boundary-value problems and continuous coefficients

Also uses Ordinary derivatives and the product rule, Exponential functions, sine and cosine and Complex conjugate roots and two by two determinants.

Theory part 91Practice part 92

complementary and particular solutions, undetermined coefficients, resonance adjustments, variation of parameters and coefficient-continuity assumptions

Also uses Second-order homogeneous linear equations and Wronskians and Product rule, integration and trigonometric identities.

Theory part 93Practice part 94

spring-mass system, simple harmonic motion, damping, underdamped, critically damped, overdamped, forcing, resonance, RLC circuits and transient and steady response

Also uses Second-order constant-coefficient linear equations, Trigonometry and initial conditions and Force, displacement and elementary circuit quantities introduced in the lesson.

Theory part 95Practice part 96

power series ansatz, recurrence relations, ordinary and singular points, convergence, Frobenius idea and Bessel equation and special functions

Also uses Power series and convergence, Second-order linear differential equations and Complex exponentials.

Theory part 97Practice part 98

Laplace transform as an improper integral, region of convergence, Laplace transforms of derivatives, inverse Laplace transform, step functions, impulse as a generalised input, convolution and solving linear initial-value problems

Also uses Improper integrals, Linear differential equations and initial values and Integration by parts.

Theory part 99Practice part 100

first-order vector systems, matrices, eigenvalues and eigenvectors, diagonalizable systems, repeated-eigenvalue cases, matrix exponential, phase portraits and stability

Also uses Matrices and linear maps, Scalar linear differential equations and Complex numbers and exponentials.

Theory part 101Practice part 102

autonomous systems, equilibria, Jacobian linearisation, hyperbolic equilibrium classification, inconclusive nonhyperbolic cases, predator-prey models, trajectories and local versus global behaviour

Also uses Vector fields and trajectories, Jacobian matrices and Linear-system eigenvalues and stability.