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.<\/p>\n
01 (0:51) “Vectors & matrices: Intro” \u00a0“Your journey is not a short one”. \u00a0To learn “calculus, the mathematics of the nonlinear”, prepare with “the mathematics of the linear”.<\/p>\n
01 (3:25) “Prologue”\u00a0definition of multivariate (multiple inputs and multiple outputs). \u00a0Asks\u00a0why do we care? \u00a0(graphs of surfaces, arbitrary dimensions). \u00a0linear algebra will “help us do calculus”. \u00a0“Calculus involves approximating nonlinear functions with linear functions”, so start with “the mathematics of linear multivariable functions”.<\/p>\n
Why learn about vectors and matrices? \u00a0machine learning, statistics, information from data, geometry (distance, area, volume), determinants will help calculate areas and volumes.<\/p>\n
algebra + work + fun.<\/p>\n
01.01.00 (0:35) “Lines & planes: intro”.<\/p>\n
01.01.01 (3:50) “Formulae for lines & planes”. \u00a0Lines in the plane: y = mx + b, (y-y0) = m(x-x0) (point slope form), x\/a + y\/b = 1 (intercept form).<\/p>\n
Example: a line passing through a point with a particular slope; a line passing through two points.<\/p>\n
Orthogonal: (the orthogonal slope is the negative reciprocal).<\/p>\n
01.01.02 (3:33) \u00a0“Implicit planes in 3d”. \u00a0These are analogous to lines in the plane. \u00a0n1(x-x0) + n2(y-y0)+n3(z-z0) = 0 (point slope form). \u00a0x\/a + y\/b + z\/c = 1 (intercept form) –> he says intercept form shows up in economics.<\/p>\n
example: equation of a plane passing through a point and parallel to another plane.<\/p>\n
01.01.03 (5:21) “parameterized lines in 3d”. \u00a0add an “auxiliary variable” (a parameter). \u00a0x(t) = 3r-5, y(r) = r+3, z(r) = -4r+1. \u00a0The name of the parameter doesn’t matter, and shifts or changes to the parameter that happens in all three equations doesn’t matter.<\/p>\n
Examples: find a line through two points in 3-space; find a line orthogonal to a plane and through a specific point.<\/p>\n
What will happen in higher dimensions? \u00a0“hyperplanes”, “subspaces”.<\/p>\n
01.01.04 (1:52) “Bonus! Machine learning”. \u00a0hyperplanes come up in analyzing data. \u00a0A space of images. \u00a0A “support vector machine is a hyperplane that optimally separates two types of data points”. \u00a0The video illustrates how flat planes usually won’t cut it to separate two datasets, so we’ll need nonlinear ideas (i.e. calculus).<\/p>\n
01.01 (0:25) “The big picture”: lines and planes are the start of the story!<\/p>\n","protected":false},"excerpt":{"rendered":"
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 “the mathematics of the linear”. […]<\/p>\n","protected":false},"author":8032,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_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}},"categories":[1010,157885],"tags":[],"class_list":["post-186","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-30","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":186,"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":58,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2017\/08\/25\/vector-calculus-earlier-in-the-semester\/","url_meta":{"origin":186,"position":1},"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":193,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-3-and-4\/","url_meta":{"origin":186,"position":2},"title":"Notes on Calculus Blue Volume 1, Chapters 3, 4, 5, 6","author":"siams","date":"22 July 2019","format":false,"excerpt":"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\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":186,"position":3},"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":32,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2017\/08\/25\/reading-student-learning-objectives-and-mathematics-teaching\/","url_meta":{"origin":186,"position":4},"title":"Reading “Student learning objectives and mathematics teaching”","author":"siams","date":"25 August 2017","format":false,"excerpt":"I am working to write learning objectives for multivariable calculus this Fall. This article helped distinguish between overarching goals (that students are able to fit the math in the course into a greater understanding of math and of the world) and the learning objectives, of what I hope students will\u2026","rel":"","context":"In "Learning and teaching"","block_context":{"text":"Learning and teaching","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/learning-and-teaching\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":65,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2017\/10\/04\/other-versions-of-multivariable\/","url_meta":{"origin":186,"position":5},"title":"Other versions of multivariable","author":"siams","date":"4 October 2017","format":false,"excerpt":"Resources for multivariable calculus: Some challenge problems (not multivariable): https:\/\/www.math.unl.edu\/~mrammaha1\/Challenging%20problems\/Challenge-Problems.pdf Cornell is using workshops activities for their engineering students this year (2017): http:\/\/www.math.cornell.edu\/~web1920\/workshop.html Materials from Math 53 at Berkeley (2016): https:\/\/math.berkeley.edu\/~auroux\/53s16\/ Lots of past multivariable exams for Math 215 at Michigan: http:\/\/www.math.lsa.umich.edu\/courses\/215\/17exampractice\/index.html","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":[]}],"_links":{"self":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/186","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=186"}],"version-history":[{"count":4,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/186\/revisions"}],"predecessor-version":[{"id":192,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/186\/revisions\/192"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=186"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=186"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=186"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}