Lesson readers · Calculus Volume 3

Calculus Volume 3

One page per subsection. Each opens at Level 1 (the picture), goes to Level 2 (the computation, with labs you can drag) and Level 3 (why it is true). The textbook pages sit in the side panel; page links open them at the spot.

Chapter 2 · Vectors in Space

2.1 Vectors in the PlaneArrows with a length and a direction: sliding arrows, adding tip to tail, components, magnitude, unit vectors and resultant forces — with draggable vectors and answers to the section's exercises. 2.2 Vectors in Three DimensionsThree axes, eight octants, distance as Pythagoras twice, planes and spheres from one equation, and vectors in space — with 3D figures you can turn and answers to the section's exercises. 2.3 The Dot ProductMultiply two vectors and get a number that measures how much they agree: the angle between them, the right-angle test, direction cosines, projections and work — with 2D and 3D figures, four labs and answers to the section's exercises. 2.4 The Cross ProductA vector perpendicular to two others, as long as their parallelogram's area: the right-hand rule, the determinant, area, volume and torque — with 3D figures you can turn and answers to the section's exercises. 2.5 Equations of Lines and Planes in SpaceA point and a direction give a line; a point and a normal give a plane. Vector, parametric and symmetric equations, skew lines, planes through three points, distances and angles — with turnable 3D labs and answers to the section's exercises. 2.6 Quadric SurfacesSurfaces from second-degree equations: cylinders, traces, the six quadric surfaces, and reading a surface off its equation by completing the square — with surfaces you can slice and turn, and answers to the section's exercises. 2.7 Cylindrical and Spherical CoordinatesTwo more ways to name a point in space — polar coordinates plus a height, and a distance plus two angles: conversions, the surfaces each coordinate sweeps out, and latitude and longitude — with 3D figures you can turn and answers to the section's exercises.

Chapter 4 · Differentiation of Functions of Several Variables

4.1 Functions of Several VariablesOne output from two or three inputs: domains and ranges, graphs as surfaces, level curves and contour maps, vertical traces and level surfaces — with a turnable surface linked to its contour map and answers to the section's exercises. 4.2 Limits and ContinuityLimits where a point can be reached from every direction: δ discs, paths that disagree, the limit laws, boundary points, continuity and δ balls in space — with surfaces you can turn, an ε–δ game and the section's answers. 4.3 Partial DerivativesFreeze every variable but one and differentiate: slopes of a surface along x and along y, estimates from a contour map, second partials and Clairaut's theorem, the heat and wave equations — with a slicing surface, a shrinking secant and answers to the section's exercises. 4.4 Tangent Planes and Linear ApproximationsZoom in on a smooth surface and it turns flat: the tangent plane from two slopes, linear approximation, what differentiable means for two variables, and differentials — with surfaces you can turn and zoom and answers to the section's exercises. 4.5 The Chain RuleHow fast a function of several variables changes when its inputs move: one term per route through a tree diagram, the generalized chain rule, and implicit differentiation by partial derivatives — with a point moving across a contour map, a clickable tree and draggable tangent lines. 4.6 Directional Derivatives and the GradientThe slope of a surface in any direction, and the one vector that knows every such slope: directional derivatives, the gradient, steepest ascent, normals to level curves and level surfaces — with a direction dial, a gradient field, a climb on the surface and answers to the section's exercises. 4.7 Maxima/Minima ProblemsTops, bottoms and saddles of a surface: critical points, the second derivative test, and absolute extrema on a closed region — with surfaces you can turn, a draggable point that reads the discriminant, and answers to the section's exercises. 4.8 Lagrange MultipliersThe best point you are allowed to reach: where a level curve just touches the constraint, the gradients line up, ∇f = λ∇g — with a point you walk along the constraint, the systems solved step by step, and answers to the section's exercises.