Every lesson in the Calculus III slide course, in full text: 16 decks, 1815 slides.
Week 1 - Vectors & Their PropertiesThis deck distinguishes vectors from scalars and covers component form and position vectors in two and three dimensions, then sets up the space coordinate system with distance and spheres. It works through addition, scaling, magnitude, unit vectors, and the standard basis, then the dot product and its uses for angle, orthogonality, projections, and work, and the cross product in determinant form, with the right-hand rule, area, and the triple scalar product for volume. It targets the classic traps: mixing up the scalar dot product with the vector cross product, forgetting that the cross product is not commutative, skipping normalization, and reporting a scalar projection when a vector was asked for.
Week 2 - Geometry of SpaceThis deck covers lines and planes in space and the distances between them, then cylinders, the six quadric surfaces, and the cylindrical and spherical coordinate systems. It targets the confusion between a normal vector and a direction vector, the sign pattern that separates a one-sheet surface from a two-sheet one, and the absolute-value error in the distance-to-a-plane formula.
Week 3 - Vector-Valued FunctionsThis deck covers vector-valued functions and the space curves they trace: domain, limits and continuity taken componentwise, and the derivative read as a tangent and velocity vector. It then gives the product rules for dot and cross products, defines smooth curves, integrates with a vector constant, works projectile motion, and shows why a curve of constant magnitude has its position vector orthogonal to its velocity. It targets three classic errors: differentiating a dot or cross product without the product rule, treating the constant of integration as a scalar, and forgetting that the derivative is taken component by component.
Week 4 - Unit Tangent, Arc Length & CurvatureThis deck covers the unit tangent and principal unit normal vectors, the arc length of a space curve as the integral of speed, and the arc-length parameter. It then gives three curvature formulas, the radius of curvature, the osculating circle, and the split of acceleration into its tangential and normal components. It targets the classic errors: skipping normalization before finding N, integrating the wrong quantity for arc length, and reaching for the wrong curvature formula.
Week 5 - Functions of Several VariablesThis deck covers functions of two and three variables: domain, range, graphs as surfaces, level curves and level surfaces, and the limits story. It targets three misconceptions: that two agreeing paths prove a limit exists, that a boundary point can be substituted freely, and that a 0/0 form automatically means there is no limit.
Week 6 - Partial Derivatives & Chain RulesThis deck covers partial derivatives - hold the other variable constant, and read the result as a slope - then higher-order and mixed partials with Clairaut's theorem, the total differential and linear approximation, the multivariable chain rule with tree diagrams, and implicit differentiation. It targets the classic traps: forgetting to hold a variable constant, confusing the order of a mixed partial, and dropping a term by using the single-variable chain rule where there are several paths.
Week 7 - Gradient, Extrema & Lagrange MultipliersThis deck covers the gradient and directional derivatives, steepest ascent, tangent planes and normal lines, relative and absolute extrema with the Second Partials Test, and constrained optimization by Lagrange multipliers. It targets the classic traps: using a direction vector that is not a unit vector, misreading the D-test, forgetting the boundary, and dropping the constraint equation.
Week 8 - Iterated Integrals & AreaThis deck covers partial, or iterated, integration: integrating one variable while holding the other constant, evaluating iterated integrals with constant and with variable limits, finding plane area as the double integral of 1, setting limits from a sketch, and reversing the order of integration. It targets the traps of letting the outer limits contain the outer variable, reversing the order without redrawing the region, and mismatching the differentials.
Week 9 - Double Integrals & VolumeThis deck builds the double integral from Riemann sums up to volume under a surface. It covers the properties of the integral, Fubini's theorem in either order, the volume between two surfaces as top minus bottom, average value, and setting up Type I and Type II regions. It targets the classic traps: forgetting top minus bottom, blindly swapping the limits over a region that is not a rectangle, and confusing a signed integral with a true volume.
Week 10 - Change of Variables: Polar CoordinatesThis deck covers double integrals in polar coordinates and the general change of variables. It builds the area element from a polar rectangle, explains why the extra factor of r must appear, and introduces the Jacobian with polar coordinates as the special case. It targets the classic errors: dropping the r in the area element, using the wrong angle range, leaving the integrand in x and y, and omitting the absolute value of the Jacobian.
Week 11 - Triple Integrals & ApplicationsThis deck covers triple integrals over solid regions: the volume element, finding the six nested-limit orders for a given solid, and projecting onto a coordinate plane, with applications to mass, center of mass, centroids, moments, and moments of inertia. It targets the classic traps of putting an outer variable in the inner limits, forgetting the density factor when computing mass, and projecting onto the wrong plane.
Week 12 - Cylindrical & Spherical IntegralsThis deck covers triple integrals in cylindrical and spherical coordinates: what each coordinate means, how to convert between systems, and the all-important volume factors r and rho-squared sin-phi. It targets the classic traps of dropping the volume factor, swapping phi and theta, and choosing the harder coordinate system.
Week 13 - Vector Fields & Line IntegralsThis deck covers vector fields in the plane and in space, gradient fields, divergence and curl, and line integrals of both scalar fields and vector fields, the latter giving work. It targets the classic errors: dropping the speed factor in a scalar line integral, mishandling orientation, and mis-parameterizing a curve.
Week 14 - Conservative Fields & Green's TheoremThis deck covers conservative vector fields and potential functions, the cross-partial and curl tests, the Fundamental Theorem of Line Integrals and the path independence it gives, and Green's Theorem in both its circulation and flux forms, including finding area by Green's Theorem. It targets the traps of ignoring orientation and closure, claiming that a field is conservative on a domain with a hole, and dropping the extra function when rebuilding a potential.
Week 15 - Divergence & Stokes's TheoremsThis deck covers parametric surfaces and surface area, scalar surface integrals, and oriented surfaces and flux, then reaches the two capstone theorems: the Divergence Theorem of Gauss, and Stokes's Theorem. It targets the classic errors: applying the Divergence Theorem to an open surface, orienting the normal the wrong way, and confusing when to reach for Stokes and when for Divergence.
Week 16 - Comprehensive ReviewThis deck is a guided tour that ties the whole course together. It covers the three pillars - vectors and the geometry of space, multivariable differentiation, and multivariable integration - along with the family of big theorems, and gives a decision guide for choosing a coordinate system and choosing a theorem. It targets the exam-killing traps: a missing Jacobian or volume factor, the wrong coordinate system, and the wrong big theorem.
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