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