{"id":83,"date":"2018-03-03T15:15:29","date_gmt":"2018-03-03T19:15:29","guid":{"rendered":"https:\/\/blogs.harvard.edu\/siams\/?p=83"},"modified":"2018-03-03T20:12:32","modified_gmt":"2018-03-04T00:12:32","slug":"the-vector-calculus-bridge-project","status":"publish","type":"post","link":"https:\/\/archive.blogs.harvard.edu\/siams\/2018\/03\/03\/the-vector-calculus-bridge-project\/","title":{"rendered":"The vector calculus bridge project"},"content":{"rendered":"<p>Reading about &#8220;The vector calculus bridge project&#8221; (Tevian Dray and Corinne Manogue at Oregon State).<\/p>\n<p><a href=\"http:\/\/math.oregonstate.edu\/bridge\/talks\/OSU.pdf\" target=\"_blank\" rel=\"noopener noreferrer nofollow\">Click to access OSU.pdf<\/a><\/p>\n<p><a href=\"http:\/\/physics.oregonstate.edu\/~tevian\/bridge\/papers\/FEdgap.pdf\" target=\"_blank\" rel=\"noopener noreferrer nofollow\">Click to access FEdgap.pdf<\/a><\/p>\n<p>Their takeaways:<br \/>\n* key calculus idea: the differential (not limits)<br \/>\n* key derivative idea: rates of change (not slopes)<br \/>\n* key integral idea: total amounts (not areas\/volumes)<br \/>\n* key curves\/surfaces idea: &#8220;use what you know&#8221;? (not parameterization)<br \/>\n* key function idea: &#8220;data attached to the domain&#8221; (not graphs)<\/p>\n<p>Differences in interpretation between mathematicians and physicists example 1:<br \/>\nIf T(x,y) = k(x^2+y^2) then is T(r,\\theta) = kr^2 or T(r,\\theta) = k(r^2+\\theta^2)?\u00a0 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.<br \/>\nexample 2:<br \/>\nphysicists will use $$\\hat{\\theta}$$ to describe the direction of a magnetic field that points outwards from a wire.\u00a0 But this notation is missing from many math classes.<\/p>\n<p>Reading Dray and Manogue, &#8220;Using differentials to bridge the vector calculus gap&#8221;, The College Mathematics Journal (2003).<\/p>\n<p>Contrast the treatment of surface and line integrals in Stewart (math) and Griffiths (physics &#8211; see his electrodynamics book for a summary of calculus in 60 pages).\u00a0 They compare flux integrals over a sphere.<\/p>\n<ul>\n<li>(math) compute the normal vector via a parameterization of the surface in terms of two coordinates.\u00a0 Use the cross product.<\/li>\n<li>(physics) reason out the direction of the unit normal vector and the surface element size.\u00a0 Find the dot product between the vector field and the unit normal vector, then integrate.<\/li>\n<\/ul>\n<p>The math version was general and did not involve geometric reasoning.\u00a0 The physics version used prior knowledge of the sphere.<\/p>\n<p>Vector differentials: They recommend using &#8220;adapted basis vectors such as&#8221; $$\\hat{r}, \\hat{\\theta}, \\hat{\\phi}$$ in addition to $$\\hat{i}, \\hat{j}, \\hat{k}$$.\u00a0 To approach a paraboloid, they suggest using $$d\\vec{r}$$ in cylindrical coordinates.<\/p>\n<p>Reading Dray and Manogue, &#8220;Putting differentials back into calculus&#8221;, The College Mathematics Journal (2010).<\/p>\n<p>&#8220;The differentials of Leigniz&#8230; capture the essence of calculus&#8221;.\u00a0 Think of d as &#8220;a little bit of&#8221; and an integral symbol as &#8220;a long S, and may be called&#8230; &#8216;the sum of'&#8221;.<\/p>\n<p>Differentials are used in u-substitutions in integrals.<br \/>\nd(uv) = v du + u dv.\u00a0 This could lead to an implicit derivative (divide by du) or to relating rates (dt instead).\u00a0 &#8220;It is a statement about the relative rates&#8221;.<\/p>\n<p>&#8220;The use of differentials turns the chain rule, implicit differentiation, and related rates&#8230; into something each rather than hard&#8221;.\u00a0 See Thompson&#8217;s 100 year-old text for differentials in the context of a textbook.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Reading about &#8220;The vector calculus bridge project&#8221; (Tevian Dray and Corinne Manogue at Oregon State). Click to access OSU.pdf Click to access 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: &#8220;use [&hellip;]<\/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-83","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-1l","jetpack-related-posts":[{"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":83,"position":0},"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 &quot;Math&quot;","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":83,"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 &quot;Math&quot;","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":83,"position":2},"title":"Notes on &#8220;Calculus Blue&#8221; 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 &quot;Math&quot;","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":83,"position":3},"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 &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":188,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2019\/07\/22\/notes-on-calculus-blue-volume-1-chapter-2\/","url_meta":{"origin":83,"position":4},"title":"Notes on &#8220;Calculus Blue&#8221; 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 &quot;Math&quot;","block_context":{"text":"Math","link":"https:\/\/archive.blogs.harvard.edu\/siams\/category\/math\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":88,"url":"https:\/\/archive.blogs.harvard.edu\/siams\/2018\/03\/05\/calculus-made-easy\/","url_meta":{"origin":83,"position":5},"title":"Calculus made easy","author":"siams","date":"5 March 2018","format":false,"excerpt":"S.P. Thompson.\u00a0 Calculus made easy: Being a very-simplest introduction to those beautiful methods of reckoning which are generally called by the terrifying names of the differential calculus and the integral calculus. (1914). \u00a0 It is interesting to take a look at this 1914 approach to calculus. \u00a0","rel":"","context":"In &quot;Math&quot;","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\/83","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=83"}],"version-history":[{"count":3,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/83\/revisions"}],"predecessor-version":[{"id":95,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/posts\/83\/revisions\/95"}],"wp:attachment":[{"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/media?parent=83"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/categories?post=83"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive.blogs.harvard.edu\/siams\/wp-json\/wp\/v2\/tags?post=83"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}