Calculus III (Multivariable)

Units

1. 1

   Vectors and the geometry of space

   9 lessons

   1. Three-dimensional coordinatesLocate a point in space by three numbers, find the distance and midpoint between two points, and read x = a as a plane and (x − h)² + (y − k)² + (z − l)² = r² as a sphere.
   2. Vectors in spaceDescribe a move in space as a vector with three components, find its length, add and scale vectors, and build unit vectors, using a drone flying against the wind.
   3. The dot productCompute the dot product of two vectors in space, use it to find the angle between them, test for a right angle, and find how much of one vector lies along another, including direction angles and work.
   4. The cross productMultiply two vectors in space into a third vector perpendicular to both, read its length as an area, and use it for the twist of a wrench and the volume of a slanted box.
   5. Lines in spaceWrite the straight flight path through two waypoints as a vector, parametric and symmetric equation, test points and segments on it, and tell whether two paths are parallel, intersecting or skew.
   6. Planes in spaceWrite the equation of a plane from a point and a normal vector, or from three points using the cross product, and read its intercepts, traces, parallel planes and the angle between two planes.
   7. Distances and intersectionsFind the shortest distance from a point to a plane or a line, between parallel planes and between skew lines, and find where a line meets a plane or two planes meet, using a drone near a roof panel and a power line.
   8. Cylinders and tracesFind the trace of a surface in a plane x = k, y = k or z = k, recognise a cylinder from an equation with a missing variable, and read a surface from its traces, using slices of curved roofs.
   9. Quadric surfacesRecognise the six quadric surfaces from their equations and traces, tell them apart by the signs, and use completing the square to find the type and centre of a shifted quadric, using a cooling tower and a satellite dish.
2. 2

   Vector functions

   7 lessons

   1. Vector functions and space curvesTreat the position along a spiral staircase as a vector function r(t) of one parameter, read its curve and shadow, take limits one component at a time, and parametrise a curve where two surfaces meet.
   2. Derivatives of vector functionsDifferentiate a vector function r(t) as a limit of chord vectors, read r'(t) as the tangent vector and velocity, build the unit tangent and the tangent line, and use the product rules.
   3. Integrals of vector functionsIntegrate a vector function one component at a time, treat the constant of integration as a vector fixed by an initial condition, and recover velocity and position of a thrown ball from its acceleration.
   4. Arc length and parametrizationFind the length of a space curve as the integral of its speed, build the arc length function s(t), and reparametrize a curve by arc length so that its speed is 1.
   5. CurvatureDefine the curvature κ as the rate at which the unit tangent turns per unit of arc length, find it for a circle, the helix and a plane curve, and use κ = |r' × r''|/|r'|³.
   6. The normal and binormal vectorsBuild the unit normal N and the binormal B from T, assemble the TNB frame, write the osculating and normal planes and the osculating circle, and meet the torsion as the twist of B.
   7. Motion in spaceDescribe motion in space with the velocity, speed and acceleration vectors, find the path of a projectile under gravity, and split the acceleration into a tangential part a_T = v' and a normal part a_N = κv².
3. 3

   Partial derivatives

   12 lessons

   1. Functions of several variablesRead the height of a hill as a function f(x, y) of two numbers, see it as a surface, a contour map and a table of values, and extend the idea to a function of three variables.
   2. Limits and continuity in several variablesDecide what a function of two variables approaches as the point (x, y) closes in on a spot, show by two paths that a limit does not exist, prove one that does with a bound or polar coordinates, and test continuity.
   3. Partial derivativesMeasure how fast the hill's height changes due east and due north as a limit, find f_x and f_y by freezing the other variable, and read them as rates of change from a formula, a table and a function of three variables.
   4. Higher-order partial derivativesDifferentiate the slopes of the hill again to get the four second partials, read them as bends of the traces, use Clairaut's theorem to swap the order of the mixed ones, and test Laplace's and the wave equation.
   5. Tangent planes and linear approximationBuild the flat plane that touches the hill at a spot from the two slopes, use its height to estimate nearby heights, bound errors with the differential, and say when the plane really fits.
   6. The chain rule for several variablesFind how fast the height of the hill changes along a path by combining the two slopes with the walker's two speeds, draw the tree diagram that records the rule, and extend it to two parameters and more variables.
   7. Implicit partial derivativesFind the slope of a level curve and the slopes ∂z/∂x and ∂z/∂y of a surface given by an equation F = k, without solving for the unknown, and check the result against an explicit formula where one exists.
   8. Directional derivativesFind the slope of the hill in any direction you choose to walk, by weighting the east and north slopes with the parts of a unit vector, and apply it to a direction given by any vector and to three variables.
   9. The gradient vectorPack the two partial derivatives into the gradient ∇f = ⟨f_x, f_y⟩, use it to find the steepest direction and the steepest rate, show that it stands perpendicular to level curves, and extend it to level surfaces, tangent planes and normal lines.
   10. Local maxima, minima and saddle pointsFind the critical points of f(x, y) by solving f_x = 0 and f_y = 0 together, and classify each as a local maximum, a local minimum or a saddle point with the second derivative test D = f_xx f_yy − (f_xy)², including what to do when D = 0.
   11. Absolute extrema on a closed regionFind the largest and smallest values of f(x, y) on a closed and bounded region by listing the interior critical points and the one-variable extremes on each edge, comparing all values, and extend the idea to a box-volume word problem.
   12. Lagrange multipliersFind the largest and smallest values of f on a constraint curve g(x, y) = k by solving ∇f = λ∇g together with g = k, understand why the gradients are parallel, compare the candidate values, and use the same method for a function of three variables.
4. 4

   Multiple integrals

   10 lessons

   1. Double integrals over rectanglesEstimate the volume under a surface over a rectangle with columns, define the double integral as the limit of those sums, compute it as an iterated integral in either order, and use a table of heights for a midpoint estimate and an average value.
   2. Double integrals over general regionsDescribe a curved region of the plane with vertical or horizontal strips, set the limits of an iterated integral from a sketch, and use it for the region's area and for the volume of the block above it.
   3. Changing the order of integrationRead the region from the limits of an iterated integral, redraw it with strips cut the other way, rewrite the limits, and use the swap to evaluate integrals whose inner integral cannot be done, or to avoid splitting a region.
   4. Double integrals in polar coordinatesDescribe a disk, ring or sector by constant polar limits, see from a polar cell why dA = r dr dθ, convert a double integral to polar form, and use it for volumes, regions between curves and the Gaussian integral.
   5. Applications of double integralsUse double integrals for the mass, balance point and moment of inertia of a thin plate with varying density, for an expected value of a joint probability density, and for the area of a surface z = f(x, y).
   6. Triple integralsAdd up a quantity over a solid by filling it with tiny boxes, write the sum as three nested integrals, set the limits from the solid's floor, roof and shadow, and compute volume and mass in more than one order.
   7. Cylindrical coordinatesName points of space by (r, θ, z), convert both ways, see why the volume element carries a factor r, and convert and evaluate triple integrals over solids with a round axis.
   8. Spherical coordinatesName points of space by (ρ, φ, θ) with φ measured from the positive z-axis, convert both ways, see why the volume element is ρ² sin φ dρ dφ dθ, and integrate over balls and cones.
   9. Applications of triple integralsUse triple integrals for the mass, balance point and moment of inertia of a solid with density, for the average value of a function over a solid, and choose the coordinate system that suits the solid.
   10. Change of variables and the JacobianUse a transformation to turn a tilted or curved region into a simple one, find the Jacobian as the local area stretch factor, and write ∬_R f dA as ∬_S f |J| du dv, including the ellipse, polar and spherical cases.
5. 5

   Vector calculus

   13 lessons

   1. Vector fieldsUnderstand a vector field as an arrow attached to every point, read and evaluate F = ⟨P, Q⟩ for the swirl ⟨−y, x⟩, follow flow lines, meet gradient fields and show the swirl is not one, and extend to fields in space such as the inverse-square field.
   2. Line integrals of scalar functionsAdd up a scalar function f along a curve C as ∫_C f ds, compute it with ds = |r′(t)| dt, see that it ignores the pace and the direction of travel, and extend it to pieces, closed curves and wires in space.
   3. Line integrals of vector fieldsCompute the work ∫_C F · dr = ∫ F(r(t)) · r′(t) dt of a force field along a path, see how orientation changes the sign and how the route changes the answer, and use the form P dx + Q dy and the circulation round a closed curve.
   4. The fundamental theorem for line integralsProve and use ∫_C ∇f · dr = f(B) − f(A), see that the work of a gradient field depends only on the end points and is zero round any closed curve, and contrast it with the swirl ⟨−y, x⟩, which is not a gradient field.
   5. Conservative fields and potential functionsDecide whether a vector field is the gradient of a function, using the cross partials, find that function, and use it to evaluate work from the two endpoints alone, while knowing why the test needs a region with no holes.
   6. Green's theoremTurn the circulation of a field around a closed curve in the plane into a double integral over the region inside, keep the orientation and the conditions right, and use the theorem to evaluate hard line integrals and to find areas.
   7. Curl and divergenceMeasure how a field spins and how it spreads out at a point, write both with the del operator, use the two identities that follow, and meet Green's theorem in its circulation and flux forms.
   8. Parametric surfaces and their areasDescribe a curved surface with two parameters, r(u, v), read its grid curves, find tangent vectors, the normal and the tangent plane at a point, and compute its area with |r_u × r_v|, including the sphere, graphs and surfaces of revolution.
   9. Surface integrals of scalar functionsAdd a quantity such as density over a curved surface with ∬_S f dS = ∬_D f(r(u, v)) |r_u × r_v| dA, use the graph form dS = √(1 + g_x² + g_y²) dA, and find the mass and the average value of a thin shell.
   10. Surface integrals of vector fields (flux)Measure how much a flowing field crosses a surface per second with ∬_S F · n dS = ∬_D F(r(u, v)) · (r_u × r_v) dA, choose and respect the orientation of the surface, and find the flux of F = ⟨x, y, z⟩ through a sphere.
   11. Stokes' theoremState and use Stokes' theorem, that the circulation of a field round the rim of a surface equals the flux of its curl through the surface, orient the rim and the surface with the right-hand rule, choose whichever surface is easiest, and see Green's theorem as the flat case.
   12. The divergence theoremState and use the divergence theorem, that the flux of a field out of a closed surface equals the integral of its divergence over the solid inside, read divergence as a source strength, and close an open surface to turn a hard flux into an easy triple integral.
   13. The great theorems side by sideSee the fundamental theorem of calculus, the fundamental theorem for line integrals, Green's theorem, Stokes' theorem and the divergence theorem as one pattern, choose the right one by what is integrated over what, and check an answer by a second route.