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