Category Archives: Multivariable

Notes on Calculus Blue Volume 1, Chapters 3, 4, 5, 6

More of Calculus Blue by Prof Ghrist Math.

Chapter 3 is on coordinates and distance (see section 12.1 of the 6th edition of Hughes-Hallett).

01.03 (0:36) “Coordinates: intro”.  Review coordinates and see it in data.

01.03.01 (2:16) “coordinates & many dimensions”.  from curves and surfaces we’ll head on.  plane, then 3-space.  what next?  He introduces R-n.  He mentions that the coordinates may have units and that the coordinate is a point in higher dimensional space.  But why care?  (see the next examples).

01.03.02 (2:33) “Example – robot kinematics”.  robot arm with a bunch of joint angles.  There is a configuration space.  He shows a video: the tip of a robot arm is tracing a path in 3-space.

01.03.03 (1:42) “Example – wireless signals & localization”.  the phone can sense a bunch of wireless signals at once and multiple might be nonzero, so the phone has a position in signal space.  the dimension is the number of wireless routers in the building.

01.03.04 (2:04) “Example – customer preferences & profiles”.  create a preference space of how customers feel about a bunch of products.  You’ll want to cluster (group) this dataset.

01.03.05 (7:27) “Distances via coordinates”.  we need to built up algebra and geometry.  start with distance.

Examples: (1) distance from point to line (where the line is parameterized), so the shortest distance; this could be done in single variable calculus: he foreshadows that we’ll learn a better way.  (2) configuration space for four objects on a chess board, so 8 coordinates, and take a distance in that space.  (3) maximal distance between two points in a unit ball in 49-dimensional space: it is 2.  (4) how about a 49-cube?  it is 7, which is weird.

we still need more tools (not quite time for calculus).

01.03 (0:36) “the big picture”.  higher dimensional spaces exist in systems and in data.

Total time: 17:14

Chapter 4 is an intro to vectors (see sections 13.1 and 13.2 of the 6th edition of Hughes-Hallett).

01.04 (0:38) “Vectors: intro”.  This is a tool for organizing variables or data.

01.04.01 (2:38) “Vector components”.  one way to think of them is as a difference between two points.  Another is as two objects that can be added and rescaled.  We’ll work with a coordinate system and interpret vectors as movement in that space.  Stack the components vertically in vectors.  Use an underline to denote a vector.

01.04.02 (6:32) “Basic vector algebra”.  algebra: addition, rescaling, using components and acting term by term.  properties: commutative, identity, subtraction.  geometry: concatenation, so draw u, then v, then the sum.  Define the length of a vector.  Things to prove: triangle inequality, and a couple of others.  Lines and planes can be nicely parameterized using vectors. 1d line, 1 parameter, 1 vector.  2d plane, 2 parameters, 2 vectors.  Nice animation of how the two vector are used to parameterize the plane.

01.04.03 (3:52) “Standard basis vectors”.  vec i, vec j, vec k are introduced, as are vec e_k.

example: (1) take a vector and write it as a linear combination of the vec e_k vectors.  (2) do the same for a vector in 3d using vec i, vec j, vec k.  (3) Take the length of a vector sum.

01.04.04 (1:53) “Caveat & a foreshadowing of fields”.  vectors are actually independent of how you represent them.  where’s the calculus?  we need more background!  At some point, though, we’ll learn a calculus for fields of vectors (foreshadowing of vector fields).

01.04 (0:26) “The big picture”.  Vectors carry both algebraic and geometric data.  This was our intro to them.

Total time: 16:00

Chapter 5 is on dot products (so section 13.3 of the 6th edition of Hughes-Hallett)

01.05 (0:41) “the dot product: intro”.  good data structure: geometry and algebra.

01.05.01 (1:23) “definition of the dot product”.  define it.  properties: commutative, dot with zero is zero, dot product with itself is length.

01.05.02 (3:27) “dot products & orthogonality”.  the angle between two vectors is well-defined.  memorize the geometric definition of the dot product.  use dot products to detect orthogonality.

example: (1) find an angle between two vectors with four components.  (2) can create a pair of vectors that are orthogonal.  (3) the standard basis vectors are all mutually orthogonal.

01.05.03 (3:32) “dot products as orthogonal projection”.  projected length is an important interpretation of the dot product (oriented, projected, length along an axis).  Really great animation / visualization for this!

example: (1) find the component of one vector in the direction of another.

01.05.04 (2:48) “hyperplanes & machine learning”.  use the dot product to make sense of our implicit equations for lines and planes.  Another nice animation / visualization.  hyperplanes (a “support vector machine”) separate two types of data.  with a “normal vector” to the plane you can tell which side of the plane a data point is on.

01.05.05 (1:54) “dot products and compatibility”.  love: create a preference space with a bunch of opinions.  make two vectors and then take a dot product.  a large positive dot product means two people like similar things.

01.05.06 (1:36) “foreshadowing of Fourier”.  additional math: you could learn to think of functions as infinite-dimensional vectors.  black and white foreshadowing video.

01.05 (0:27) “the big picture”.  the dot product, with algebra, geometry and applications!

Total time:  15:48

Chapter 6 is on cross products (section 13.4 of the 6th edition of Hughes-Hallett).

01.06 (0:41) “The cross product: intro”.  two more products (unique to 3d).

01.06.01 (5:35) “definition of the cross product”.  (only in 3d).  he defines it.  properties.  anti-commutativity, vec u cross vec 0 = vec 0.  cross product with itself is zero.  Look at the geometric meaning of that anti-commutativity.  The cross-product is orthogonal to both of its factors (he proves this).  Then he shows a visualization and defines the right hand rule.  The illustration involved a spinning mill.  🙂

examples: (1) three points, PQR, and find the equation of a plane by finding an orthogonal vector and put it into the formula for a plane.

01.06.02 (2:20) “computing cross products in the standard basis”.  How to remember the cross-product formula?  Use the standard basis vectors, then draw the cyclic diagram

examples: (1) find the cross product by using the standard basis vectors and expanding.

01.06.03 (2:45) “length of the cross product”.  geometric formula for the length of the cross product (a geometry problem).  the vector isn’t just orthogonal to the vec u vec u plane; its length also has a meaning.  use the cross product for a simple formula for a point to a line and the dot product for a formula from the point to a plane.

01.06.04 (3:53) “the scalar triple product”.  this product takes in 3 vectors and returns a scalar.  he gives the algebraic definition.  then he shows the cyclic repeat (5 columns) with diagonal slices for putting together the structure.  properties: there’s a cyclic permutation, anti-symmetry, and geometric meaning.  It’s the volume of a parallelepiped.

01.06.05 (1:12) “bonus: octonians”.  how about a way to multiply vectors together in another dimension?  the octonions work in the 7th dimension.  you can look them up…

01.06 (0:31) “the big picture”.  new products: cross product and scalar triple product.

Total time: 16:57

Notes on “Calculus Blue” Volume 1, Chapter 2

More notes on the Calculus Blue Multivariable Volume 1 videos on YouTube by “Prof Ghrist Math”.

Chapter 2 introduces curves in the plane and surfaces in 3-space with implicit and parametric definitions for curves in the plane and for surfaces in 3-space.  They also introduce the names and images for all of the quadratic surfaces (so we’ve left the linear world in this chapter).  This is a foreshadowing of the process of the course (start with the linear and then move to the nonlinear).

01.02.00 (0:35) “Curves and surfaces: Intro”

01.02.01 (5:10) “Implicit and parametric curves and surfaces”.  Two ways to define curves.  Implicitly or parametrically.  Implicit: “the solutions to an equation yields a curve in the plane”.  Parametric: “specifying coordinates as a function of a parameter”.  We’ll want to move between these representations.  Surfaces in 3d can also be expressed implicitly or parametrically (requires 2 parameters).

Examples: parameterization for a surface expressed implicitly (he shows the conversion where you set x = s, y = t, and then express z); go from a parameterization to an implicit equation (the example has a square root so he suggests caution).

01.02.02 (0:29) “Some examples please…?”.  it’s important to learn the quadratic surfaces.

01.02.03 (5:33) “Examples of quadratic surfaces”.  start with the sphere, then modify to get the ellipsoids, change a sign to get a hyperboloid, with two negatives signs get a 2-sheeted hyperboloid, then an elliptic paraboloid, hyperbolic paraboloids, cones (degenerate hyperbola), cylinders.

Then the narrator provides a reassurance: this isn’t about memorizing these or drawing pictures of these; it’s just worth recognizing them.  Then the narrator mentions these won’t come back for a while and names surface integrals as an application.  These are in there as a motivation to build up geometry and algebra skills.

01.02 (0:26) “The big picture”.  Previous chapter was lines and planes.  This one was curves and surfaces.  Progression from linear to nonlinear is what will happen in the course.

Notes on “Calculus Blue” Volume 1, Chapter 1

These notes are on the Calculus Blue videos by Ghrist on YouTube.  He emphasizes that the math will involve substantial (and worthwhile) work, which I really appreciate.

01 (0:51) “Vectors & matrices: Intro”  “Your journey is not a short one”.  To learn “calculus, the mathematics of the nonlinear”, prepare with “the mathematics of the linear”.

01 (3:25) “Prologue” definition of multivariate (multiple inputs and multiple outputs).  Asks why do we care?  (graphs of surfaces, arbitrary dimensions).  linear algebra will “help us do calculus”.  “Calculus involves approximating nonlinear functions with linear functions”, so start with “the mathematics of linear multivariable functions”.

Why learn about vectors and matrices?  machine learning, statistics, information from data, geometry (distance, area, volume), determinants will help calculate areas and volumes.

algebra + work + fun.

01.01.00 (0:35) “Lines & planes: intro”.

01.01.01 (3:50) “Formulae for lines & planes”.  Lines in the plane: y = mx + b, (y-y0) = m(x-x0) (point slope form), x/a + y/b = 1 (intercept form).

Example: a line passing through a point with a particular slope; a line passing through two points.

Orthogonal: (the orthogonal slope is the negative reciprocal).

01.01.02 (3:33)  “Implicit planes in 3d”.  These are analogous to lines in the plane.  n1(x-x0) + n2(y-y0)+n3(z-z0) = 0 (point slope form).  x/a + y/b + z/c = 1 (intercept form) –> he says intercept form shows up in economics.

example: equation of a plane passing through a point and parallel to another plane.

01.01.03 (5:21) “parameterized lines in 3d”.  add an “auxiliary variable” (a parameter).  x(t) = 3r-5, y(r) = r+3, z(r) = -4r+1.  The name of the parameter doesn’t matter, and shifts or changes to the parameter that happens in all three equations doesn’t matter.

Examples: find a line through two points in 3-space; find a line orthogonal to a plane and through a specific point.

What will happen in higher dimensions?  “hyperplanes”, “subspaces”.

01.01.04 (1:52) “Bonus! Machine learning”.  hyperplanes come up in analyzing data.  A space of images.  A “support vector machine is a hyperplane that optimally separates two types of data points”.  The video illustrates how flat planes usually won’t cut it to separate two datasets, so we’ll need nonlinear ideas (i.e. calculus).

01.01 (0:25) “The big picture”: lines and planes are the start of the story!

Hughes-Hallett et al Chapter 11: Differential equations

  • 11.1: What is a differential equation?
    • Starts with an example: what sets the rate at which a person learns a new task?  Defines a diff eq and a solution to a diff eq.
    • Defines order of a diff eq.  Example 1 is showing a function is not a solution to a diff eq.
    • There is just one example and it is modest.
  • 11.2: Slope fields
    • Introduce the idea of a slope field.  Example 1: compare a slope field and a solution curve.  Example 2: guess solution form by looking at a slope field.  Example 3: sketch solution curves with given initial conditions on a slope field.
    • Existence and uniqueness is introduced briefly.
  • 11.3: Euler’s method
    • Present the Euler’s method algorithm.  Example 1: use Euler’s method for a simple example.  Example 2: change the step size.  Example 3: approximate four points on a solution curve.
    • Accuracy of Euler’s method.  Introduce the idea of error.
  • 11.4: Separation of variables
    • Present the method (including the differential on each side of the equation).  Example 1: a simple separation problem.  Example 2: another simple linear problem.  Example 3: a third linear case.
  • 11.5: Growth and decay
  • 11.6: Applications and modeling
  • 11.7: The logistic model
  • 11.8: Systems of differential equations
  • 11.9: Analyzing the phase plane
  • Projects

Hughes-Hallett et al Chapter 8: Using the definite integral

For the course “Integrating and Approximating” our focus will be on multivariate integration, vector calculus, and differential equations.  In the past, I’ve used a number of texts for Multivariable, but appreciate the four-fold perspective (tables, graphs, formulas, words) that is used in Hughes-Hallett et al.

A few chapters of single variable portion of the text are particularly relevant, so I’ll summarize them via blog posts.

  • 8.1 Areas and volumes
    • Find area via horizontal slides: Example 1 is a triangle, where horizontal slices are integrated to give the area.  Example 2 is a half-disk via horizontal slices.  Introducing (or reviewing) horizontal slices is a good idea before moving to 2d.
    • Find volume via slices that are disks or squares: Example 3 is volume of a solid cone.  Vertical slices are a weird shape but horizontal ones are coins.  Example 4 is a half-ball via circular slices.  Example 5 is a pyramid, which has square slices.
    • We could set up expressions for these areas using either single or double integrals, and expressions for these volumes using single, double, or triple integrals.
  • 8.2 Applications to geometry
    • Find volume via slices that are disks or squares: Examples 1, 2, 3 are volumes made up of thickened disks or pieces of disks but that have more irregular shapes than above (think of a turned banister with varying radius).  Example 4 is an interesting and complicated shape where the cross sections are known to be squares.
    • Find arclength: Examples 5 and 6 are arclength, including with a parametric curve in 2d.
    • Arclength is worth doing in 2d (along with a parameterized curve in 2d) before returning to it in 3d: I can check whether students saw this in their Calc II course.  In addition, thinking through Example 4 would be worthwhile for working with the geometry of finding volumes.
  • 8.3 Area and arc length in polar coordinates
    • Introduce polar coordinates, including their non-uniqueness: Example 1 is translating between polar and Cartesian coordinate systems.  Example 2 is giving different polar coordinates for a single point.  Examples 3 and 4 are about graphing a curve given in polar, and translating equations between polar and Cartesian.
    • Introduce roses and limacons: Example 5 is two different roses and Example 6 is two different limacons.
    • Use inequalities to describe regions.  Example 7 is describing an filled annulus in polar and Example 8 is describing a pizza slice.
    • Find the area of a region described in polar by slicing into circular sectors: Example 9 is the area inside a limacon and Example 10 the area inside a petal of a rose graph.
    • For a curve r = f(theta), find the slope.  Example 11 uses the formula for the slope of a parametric curve to find the slope.
    • For a curve r = f(theta), find the arclength.  Example 12 is finding the arclength of one petal of a rose graph.
    • This section has a lot in it.  The comprehensive intro to polar, including area, slope, and arclength, is worthwhile.
  • 8.4 Density and center of mass
    • Finding a total quantity using a density function: Example 1 is population density along the Mass Turnpike.  Examples 2 and 3 are the mass of a solid-cylindrical column of air, and Example 4 is another population, but over a disk rather than along a line.
    • Find the center of mass or balance point.  This involves defining displacement and moment.  Example 5 is a center of mass for children on a seesaw.  Example 6 works with a definition generalized to a number of point masses and Example 7 is with a continuous mass density.
    • Find the center of mass for a 2d or 3d region.  Example 8 is an isosceles triangle and Example 9 a solid hemisphere.
    • The definition of a density function, and practice with a density function is important.
  • 8.5 Applications to physics
    • Work done by a force.  Example 1 is working with the definition.  Example 2 is the work to compress a spring.  Examples 4, 5, 6 are the work done by lifting a book, pumping oil to fill a tank, or building a pyramid.
    • Use pressure to calculate force.  Example 7 is for a sunken ship and Example 8 is for the Hoover Dam.
    • Pressure and work are both confusing topics in vector calculus so introducing them in the single variable context seems helpful.
  • 8.6 Applications to economics
    • Present and future value of money: Example 1 is comparing a lump sum lottery payment vs installments.  Example 2 is looking at the value of an income stream.
    • Supply and demand curves and consumer vs producer surplus (no associated examples)
    • This doesn’t feed into any applications that I usually present in multivariable.
  • 8.7 Distribution functions
    • Introduces a histogram (rather than using a count, they use one that is normalized, so that the total area is 1).  This leads to defining a probability density function.  Example 1 is estimating totals from a histogram, Example 2 is approximating a density function.
    • The cumulative distribution is also defined (no associated examples)
    • Introducing probability is worthwhile because most students take probability at some point, and it often isn’t introduced in a single variable course.
  • 8.8 Probability, mean, and median
    • Find probability using a probability density function.  Example 1: use a pdf to identify a probability.
    • Define median.  Example 2: use a pdf to find the median age of the US population.
    • Define mean.  Example 3: use the pdf to find the mean age.
    • Normal distribution.  Example 4: given a normal distribution with a given mean and stdev, find some probabilities.
    • These are all worth defining, and doing them in 1d is probably simpler than in 2d.
  • Projects
    • Flux of fluid from a capillary
    • Testing for kidney disease
    • Volume enclosed by crossing cylinders
    • Length of cable on the Golden Gate Bridge
    • Surface area
    • Maxwell’s distribution of molecular velocities

The vector calculus bridge project

Reading about “The vector calculus bridge project” (Tevian Dray and Corinne Manogue at Oregon State).……

Their takeaways:
* key calculus idea: the differential (not limits)
* key derivative idea: rates of change (not slopes)
* key integral idea: total amounts (not areas/volumes)
* key curves/surfaces idea: “use what you know”? (not parameterization)
* key function idea: “data attached to the domain” (not graphs)

Differences in interpretation between mathematicians and physicists example 1:
If T(x,y) = k(x^2+y^2) then is T(r,\theta) = kr^2 or T(r,\theta) = k(r^2+\theta^2)?  The first option is thinking about the meaning of the function in the world; the second is thinking about a function of two variables as an input-output relationship.
example 2:
physicists will use $$\hat{\theta}$$ to describe the direction of a magnetic field that points outwards from a wire.  But this notation is missing from many math classes.

Reading Dray and Manogue, “Using differentials to bridge the vector calculus gap”, The College Mathematics Journal (2003).

Contrast the treatment of surface and line integrals in Stewart (math) and Griffiths (physics – see his electrodynamics book for a summary of calculus in 60 pages).  They compare flux integrals over a sphere.

  • (math) compute the normal vector via a parameterization of the surface in terms of two coordinates.  Use the cross product.
  • (physics) reason out the direction of the unit normal vector and the surface element size.  Find the dot product between the vector field and the unit normal vector, then integrate.

The math version was general and did not involve geometric reasoning.  The physics version used prior knowledge of the sphere.

Vector differentials: They recommend using “adapted basis vectors such as” $$\hat{r}, \hat{\theta}, \hat{\phi}$$ in addition to $$\hat{i}, \hat{j}, \hat{k}$$.  To approach a paraboloid, they suggest using $$d\vec{r}$$ in cylindrical coordinates.

Reading Dray and Manogue, “Putting differentials back into calculus”, The College Mathematics Journal (2010).

“The differentials of Leigniz… capture the essence of calculus”.  Think of d as “a little bit of” and an integral symbol as “a long S, and may be called… ‘the sum of'”.

Differentials are used in u-substitutions in integrals.
d(uv) = v du + u dv.  This could lead to an implicit derivative (divide by du) or to relating rates (dt instead).  “It is a statement about the relative rates”.

“The use of differentials turns the chain rule, implicit differentiation, and related rates… into something each rather than hard”.  See Thompson’s 100 year-old text for differentials in the context of a textbook.


Other versions of multivariable

Resources for multivariable calculus:

Vector calculus earlier in the semester?

Flux is a particularly central scientific and mathematically idea that appears in the context of a multivariable calculus course.  Given a velocity vector field and a surface, the flux of the vector field through the surface tells us the rate at which fluid is flowing through the surface.  This leads to important ideas of balance.  If material is flowing out of a closed surface, then either the amount of material in the enclosed region is changing in time, or there is some source of material sitting within the region.

Flux it is often covered towards the end of a multivariable course, in the vector calculus portion.  In the article “Early Vector Calculus: A Path Through Multivariable Calculus” by Robert Robertson, he discusses a path through the course that makes the idea of flux more central.  To work with flux, and the related divergence theorem, students need to have been exposed to partial derivatives, vectors and the vector dot product, solid regions and their bounding surfaces, triple integrals, surface integrals, and vector fields.  For surface integrals, the cross product is useful for constructing area patches on a surface (with corresponding normal vectors) that are taken to an infinitesimal limit to construct a flux integral.

For a solid region sitting in 3-space, the divergence theorem relates the the triple integral of the divergence of a vector field over a region to the flux through the surface.  Making sense of the triple integral, being able to set it up, and being able to compute the integrand, requires familiarity with vector fields, partial derivatives, solid regions, and triple integration.  This could be done in Cartesian coordinates at first.  Making sense of the flux integral over a surface requires the notion of a surface and of an area element with a surface normal vector.  This ties into the idea of tangent planes with their normal vectors.  It also draws on knowledge of double integration.

The divergence theorem is also used over a region in 2-space with a closed curve for a boundary.  Working with it in this context requires familiarity with double integrals, vector fields, partial derivatives, and boundaries.  In addition, working with the flux across a closed curve requires familiarity with parameterizing curves, computing line integrals, and finding tangent and normal vectors to curves.

Topics often taught prior to the divergence theorem that do not obviously have immediate relevance to the theorem include directional derivatives, the chain rule, and optimization.  Further study of line integrals does make good use of background in directional derivatives and the chain rule, however the specific line integral needed for the 2D divergence theorem can perhaps be explicated without these ideas.

I often include a small amount of probability in the integration section of the course in addition to covering optimization.  It is true that these two topic areas sit somewhat to the side of other content in the course and could potentially be taught later in the course.  This is an intriguing suggestion that perhaps deserves further consideration.


Robertson, Robert L. “Early Vector Calculus: A Path Through Multivariable Calculus.” PRIMUS 23, no. 2 (2013): 133-140.