More of Calculus Blue by Prof Ghrist Math.<\/p>\n
Chapter 3 is on coordinates and distance (see section 12.1 of the 6th edition of Hughes-Hallett).<\/p>\n
01.03 (0:36) “Coordinates: intro”. \u00a0Review coordinates and see it in data.<\/p>\n
01.03.01 (2:16) “coordinates & many dimensions”. \u00a0from curves and surfaces we’ll head on. \u00a0plane, then 3-space. \u00a0what next? \u00a0He introduces R-n. \u00a0He mentions that the coordinates may have units and that the coordinate is a point in higher dimensional space. \u00a0But why care? \u00a0(see the next examples).<\/p>\n
01.03.02 (2:33) “Example – robot kinematics”. \u00a0robot arm with a bunch of joint angles. \u00a0There is a configuration space. \u00a0He shows a video: the tip of a robot arm is tracing a path in 3-space.<\/p>\n
01.03.03 (1:42) “Example – wireless signals & localization”. \u00a0the phone can sense a bunch of wireless signals at once and multiple might be nonzero, so the phone has a position in signal space. \u00a0the dimension is the number of wireless routers in the building.<\/p>\n
01.03.04 (2:04) “Example – customer preferences & profiles”. \u00a0create a preference space of how customers feel about a bunch of products. \u00a0You’ll want to cluster (group) this dataset.<\/p>\n
01.03.05 (7:27) “Distances via coordinates”. \u00a0we need to built up algebra and geometry. \u00a0start with distance.<\/p>\n
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. \u00a0(2) configuration space for four objects on a chess board, so 8 coordinates, and take a distance in that space. \u00a0(3) maximal distance between two points in a unit ball in 49-dimensional space: it is 2. \u00a0(4) how about a 49-cube? \u00a0it is 7, which is weird.<\/p>\n
we still need more tools (not quite time for calculus).<\/p>\n
01.03 (0:36) “the big picture”. \u00a0higher dimensional spaces exist in systems and in data.<\/p>\n
Total time: 17:14<\/p>\n
Chapter 4 is an intro to vectors (see sections 13.1 and 13.2 of the 6th edition of Hughes-Hallett).<\/p>\n
01.04 (0:38) “Vectors: intro”. \u00a0This is a tool for organizing variables or data.<\/p>\n
01.04.01 (2:38) “Vector components”. \u00a0one way to think of them is as a difference between two points. \u00a0Another is as two objects that can be added and rescaled. \u00a0We’ll work with a coordinate system and interpret vectors as movement in that space. \u00a0Stack the components vertically in vectors. \u00a0Use an underline to denote a vector.<\/p>\n
01.04.02 (6:32) “Basic vector algebra”. \u00a0algebra: addition, rescaling, using components and acting term by term. \u00a0properties: commutative, identity, subtraction. \u00a0geometry: concatenation, so draw u, then v, then the sum. \u00a0Define the length of a vector. \u00a0Things to prove: triangle inequality, and a couple of others. \u00a0Lines and planes can be nicely parameterized using vectors. 1d line, 1 parameter, 1 vector. \u00a02d plane, 2 parameters, 2 vectors. \u00a0Nice animation of how the two vector are used to parameterize the plane.<\/p>\n
01.04.03 (3:52) “Standard basis vectors”. \u00a0vec i, vec j, vec k are introduced, as are vec e_k.<\/p>\n
example: (1) take a vector and write it as a linear combination of the vec e_k vectors. \u00a0(2) do the same for a vector in 3d using vec i, vec j, vec k. \u00a0(3) Take the length of a vector sum.<\/p>\n
01.04.04 (1:53) “Caveat & a foreshadowing of fields”. \u00a0vectors are actually independent of how you represent them. \u00a0where’s the calculus? \u00a0we need more background! \u00a0At some point, though, we’ll learn a calculus for fields of vectors (foreshadowing of vector fields).<\/p>\n
01.04 (0:26) “The big picture”. \u00a0Vectors carry both algebraic and geometric data. \u00a0This was our intro to them.<\/p>\n
Total time: 16:00<\/p>\n
Chapter 5 is on dot products (so section 13.3 of the 6th edition of Hughes-Hallett)<\/p>\n
01.05 (0:41) “the dot product: intro”. \u00a0good data structure: geometry and algebra.<\/p>\n
01.05.01 (1:23) “definition of the dot product”. \u00a0define it. \u00a0properties: commutative, dot with zero is zero, dot product with itself is length.<\/p>\n
01.05.02 (3:27) “dot products & orthogonality”. \u00a0the angle between two vectors is well-defined. \u00a0memorize the geometric definition of the dot product. \u00a0use dot products to detect orthogonality.<\/p>\n
example: (1) find an angle between two vectors with four components. \u00a0(2) can create a pair of vectors that are orthogonal. \u00a0(3) the standard basis vectors are all mutually orthogonal.<\/p>\n
01.05.03 (3:32) “dot products as orthogonal projection”. \u00a0projected length is an important interpretation of the dot product (oriented, projected, length along an axis). \u00a0Really great animation \/ visualization for this!<\/p>\n
example: (1) find the component of one vector in the direction of another.<\/p>\n
01.05.04 (2:48) “hyperplanes & machine learning”. \u00a0use the dot product to make sense of our implicit equations for lines and planes. \u00a0Another nice animation \/ visualization. \u00a0hyperplanes (a “support vector machine”) separate two types of data. \u00a0with a “normal vector” to the plane you can tell which side of the plane a data point is on.<\/p>\n
01.05.05 (1:54) “dot products and compatibility”. \u00a0love: create a preference space with a bunch of opinions. \u00a0make two vectors and then take a dot product. \u00a0a large positive dot product means two people like similar things.<\/p>\n
01.05.06 (1:36) “foreshadowing of Fourier”. \u00a0additional math: you could learn to think of functions as infinite-dimensional vectors. \u00a0black and white foreshadowing video.<\/p>\n
01.05 (0:27) “the big picture”. \u00a0the dot product, with algebra, geometry and applications!<\/p>\n
Total time: \u00a015:48<\/p>\n
Chapter 6 is on cross products (section 13.4 of the 6th edition of Hughes-Hallett).<\/p>\n
01.06 (0:41) “The cross product: intro”. \u00a0two more products (unique to 3d).<\/p>\n
01.06.01 (5:35) “definition of the cross product”. \u00a0(only in 3d). \u00a0he defines it. \u00a0properties. \u00a0anti-commutativity, vec u cross vec 0 = vec 0. \u00a0cross product with itself is zero. \u00a0Look at the geometric meaning of that anti-commutativity. \u00a0The cross-product is orthogonal to both of its factors (he proves this). \u00a0Then he shows a visualization and defines the right hand rule. \u00a0The illustration involved a spinning mill. \u00a0\ud83d\ude42<\/p>\n
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.<\/p>\n
01.06.02 (2:20) “computing cross products in the standard basis”. \u00a0How to remember the cross-product formula? \u00a0Use the standard basis vectors, then draw the cyclic diagram<\/p>\n
examples:\u00a0(1) find the cross product by using the standard basis vectors and expanding.<\/p>\n
01.06.03 (2:45) “length of the cross product”. \u00a0geometric formula for the length of the cross product (a geometry problem). \u00a0the vector isn’t just orthogonal to the vec u vec u plane; its length also has a meaning. \u00a0use 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.<\/p>\n
01.06.04 (3:53) “the scalar triple product”. \u00a0this product takes in 3 vectors and returns a scalar. \u00a0he gives the algebraic definition. \u00a0then he shows the cyclic repeat (5 columns) with diagonal slices for putting together the structure. \u00a0properties: there’s a cyclic permutation, anti-symmetry, and geometric meaning. \u00a0It’s the volume of a parallelepiped.<\/p>\n
01.06.05 (1:12) “bonus: octonians”. \u00a0how about a way to multiply vectors together in another dimension? \u00a0the octonions work in the 7th dimension. \u00a0you can look them up…<\/p>\n
01.06 (0:31) “the big picture”. \u00a0new products: cross product and scalar triple product.<\/p>\n
Total time: 16:57<\/p>\n","protected":false},"excerpt":{"rendered":"
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”. \u00a0Review coordinates and see it in data. 01.03.01 (2:16) “coordinates & many dimensions”. \u00a0from curves and surfaces we’ll head on. \u00a0plane, then 3-space. \u00a0what next? \u00a0He introduces […]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[1010,157885],"tags":[],"class_list":["post-193","post","type-post","status-publish","format-standard","hentry","category-math","category-multivariable"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p7E5LF-37","jetpack-related-posts":[{"id":188,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-2\/","url_meta":{"origin":193,"position":0},"title":"Notes on “Calculus Blue” Volume 1, Chapter 2","author":"siams","date":"22 July 2019","format":false,"excerpt":"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. \u00a0They also introduce the names and images for\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":83,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2018\/03\/03\/the-vector-calculus-bridge-project\/","url_meta":{"origin":193,"position":1},"title":"The vector calculus bridge project","author":"siams","date":"3 March 2018","format":false,"excerpt":"Reading about \"The vector calculus bridge project\" (Tevian Dray and Corinne Manogue at Oregon State). http:\/\/math.oregonstate.edu\/bridge\/talks\/OSU.pdf http:\/\/physics.oregonstate.edu\/~tevian\/bridge\/papers\/FEdgap.pdf 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\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":58,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2017\/08\/25\/vector-calculus-earlier-in-the-semester\/","url_meta":{"origin":193,"position":2},"title":"Vector calculus earlier in the semester?","author":"siams","date":"25 August 2017","format":false,"excerpt":"Flux is a particularly central scientific and mathematically idea that appears in the context of a multivariable calculus course. \u00a0Given 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. \u00a0This leads\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":134,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/17\/hughes-hallett-et-al-chapter-8-using-the-definite-integral\/","url_meta":{"origin":193,"position":3},"title":"Hughes-Hallett et al Chapter 8: Using the definite integral","author":"siams","date":"17 June 2019","format":false,"excerpt":"For the course \"Integrating and Approximating\" our focus will be on multivariate integration, vector calculus, and differential equations. \u00a0In 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\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":186,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-1\/","url_meta":{"origin":193,"position":4},"title":"Notes on “Calculus Blue” Volume 1, Chapter 1","author":"siams","date":"22 July 2019","format":false,"excerpt":"These notes are on the Calculus Blue videos by Ghrist on YouTube. \u00a0He emphasizes that the math will involve substantial (and worthwhile) work, which I really appreciate. 01 (0:51) \"Vectors & matrices: Intro\" \u00a0\"Your journey is not a short one\". \u00a0To learn \"calculus, the mathematics of the nonlinear\", prepare with\u2026","rel":"","context":"In "Math"","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":126,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/06\/11\/dynamical-systems-strogatz-chapter-3\/","url_meta":{"origin":193,"position":5},"title":"Dynamical Systems: Strogatz Chapter 3","author":"siams","date":"11 June 2019","format":false,"excerpt":"\u00a0 Section 3.0: Introduction I need to help students distinguish between parameters and variables. The beam bending example is ok, but the intuition isn't so clear. \u00a0If I back up on the load does the beam straighten (is this a supercritical pitchfork)? It would be nice to introduce an intuitive\u2026","rel":"","context":"In "Dynamical Systems"","block_context":{"text":"Dynamical Systems","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/dynamical-systems\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/193","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/users\/8032"}],"replies":[{"embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/comments?post=193"}],"version-history":[{"count":4,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/193\/revisions"}],"predecessor-version":[{"id":197,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/193\/revisions\/197"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=193"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=193"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}