{"id":535,"date":"2011-01-08T11:12:06","date_gmt":"2011-01-08T10:12:06","guid":{"rendered":"http:\/\/kmr.dialectica.se\/wp\/?page_id=535"},"modified":"2022-11-27T18:20:13","modified_gmt":"2022-11-27T17:20:13","slug":"calculus-of-several-real-variables","status":"publish","type":"page","link":"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/","title":{"rendered":"Calculus of  Several Real Variables"},"content":{"rendered":"<p>This page is a sub-page of the page on <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/\" target=\"_blank\" rel=\"noopener\">Calculus<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>The sub-pages of this page are<\/strong>:<\/p>\n<p>\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/functions-of-several-real-variables\/\" target=\"_blank\" rel=\"noopener\">Functions of Several Real Variables<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/projects\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/directional-derivative\/\" target=\"_blank\" rel=\"noopener\">Partial and Directional Derivatives<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/affine-approximation\/\" target=\"_blank\" rel=\"noopener\">Differentiation and Affine Approximation<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/projects\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/gradients\/\" target=\"_blank\" rel=\"noopener\">Gradients<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/the-chain-rule-3\/\" target=\"_blank\" rel=\"noopener\">The Chain Rule in Several Real Variables<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/projects\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/taylor-expansion-in-3d\/\" target=\"_blank\" rel=\"noopener\">Taylor Expansion in in Several Real Variables<\/a><br \/>\n\u2022 <a title=\"Integration (of functions of several real variables) at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/integration-of-functions-of-several-real-variables\/\" target=\"_blank\" rel=\"noopener\">Riemann Integration in Several Real Variables<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/partial-differential-equations\/\" target=\"_blank\" rel=\"noopener\">Partial Differential Equations<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/vector-analysis\/\" target=\"_blank\" rel=\"noopener\">Vector Analysis<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/differentials\/\" target=\"_blank\" rel=\"noopener\">Differentials<\/a><\/p>\n<p>\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/vektoranalys\/\" target=\"_blank\" rel=\"noopener\">Vektoranalys<\/a> (in Swedish)<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Related KMR-pages<\/strong>:<\/p>\n<p>\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-one-real-variable\/\" target=\"_blank\" rel=\"noopener\">Calculus of One Real Variable<\/a><br \/>\n\u2022 <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/algebra\/linear-algebra\/\" target=\"_blank\" rel=\"noopener\">Linear Algebra<\/a><br \/>\n\u2022\u00a0<a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/algebra\/matrix-algebra\/\" target=\"_blank\" rel=\"noopener\">Matrix Algebra<\/a><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Books<\/strong>:<\/p>\n<p>\u2022 <a title=\"at https:\/\/mathshistory.st-andrews.ac.uk\/Extras\" href=\"https:\/\/mathshistory.st-andrews.ac.uk\/Extras\/Courant_calculus_english\/\" target=\"_blank\" rel=\"noopener\">Differential and Integral Calculus<\/a> by <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Richard_Courant\" target=\"_blank\" rel=\"noopener\">Richard Courant<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Other related sources of information<\/strong>:<\/p>\n<p>\u2022 <a href=\"https:\/\/www.youtube.com\/playlist?list=PLjHDjmY5z0pn_p5haaVYA-Epp5Hwx1gXO\">Multivariable Calculus Theory,<\/a> by <a class=\"yt-simple-endpoint style-scope yt-formatted-string\" dir=\"auto\" spellcheck=\"false\" href=\"https:\/\/www.youtube.com\/@serpentineintegral\">Serpentine Integral<\/a> on YouTube.<br \/>\n\u2022 <a href=\"https:\/\/www.youtube.com\/watch?v=5A39Ht9Wcu0&amp;t=9s\">Understanding Lagrange Multipliers Visually<\/a>, by by <a class=\"yt-simple-endpoint style-scope yt-formatted-string\" dir=\"auto\" spellcheck=\"false\" href=\"https:\/\/www.youtube.com\/@serpentineintegral\">Serpentine Integral<\/a> on YouTube.<br \/>\n\u2022 <a title=\"Mathemaniac at YouTube\" href=\"https:\/\/www.youtube.com\/watch?v=wCZ1VEmVjVo\" target=\"_blank\" rel=\"noopener\">The right way to think about derivatives and integrals <\/a><br \/>\n\u2022 <a href=\"https:\/\/www.lamar.edu\/arts-sciences\/mathematics\/class-notes\/index.html\" target=\"_blank\" rel=\"noopener\">Lamar University \/ Department of Mathematics \/ Class notes <\/a><br \/>\n\u2022 <a title=\"at Lamar University\" href=\"https:\/\/tutorial.math.lamar.edu\/\" target=\"_blank\" rel=\"noopener\">Paul&#8217;s online notes<\/a><br \/>\n\u2022 <a title=\"at https:\/\/tutorial.math.lamar.edu\/Classes\" href=\"https:\/\/tutorial.math.lamar.edu\/Classes\/CalcII\/CalcII.aspx\" target=\"_blank\" rel=\"noopener\">Paul&#8217;s Online Notes: Calculus II <\/a><br \/>\n\u2022 <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Mathematical_analysis\" target=\"_blank\" rel=\"noopener\">Mathematical Analysis<\/a><br \/>\n\u2022 <a title=\"Methoden der matematischen Physik\" href=\"https:\/\/en.wikipedia.org\/wiki\/Methoden_der_mathematischen_Physik\" target=\"_blank\" rel=\"noopener\">Methods of Mathematical Physics<\/a>, by Richard Courant and David Hilbert<br \/>\n\u2022 <a title=\"Brilliant - Math and Science done right\" href=\"https:\/\/brilliant.org\/?utm_medium=sponsor&amp;utm_source=youtube&amp;utm_campaign=3blue1brown_180518\" target=\"_blank\" rel=\"noopener\">Brilliant &#8211; Math and science done right<\/a><br \/>\n\u2022 <a href=\"https:\/\/www.youtube.com\/playlist?list=PLSQl0a2vh4HC5feHa6Rc5c0wbRTx56nF7\" target=\"_blank\" rel=\"noopener\">Multivariable calculus<\/a>, Khan Academy on YouTube<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p>The interactive simulations on this page can be navigated with the <a href=\"http:\/\/www.pacifict.com\/FreeStuff.html\">Free Viewer<\/a><br \/>\nof the <a href=\"http:\/\/www.pacifict.com\/Home.html\">Graphing Calculator<\/a>.<\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>What they won&#8217;t teach you in calculus<\/strong> (Steven Strogatz on YouTube):<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/CfW845LNObM\" width=\"640\" height=\"360\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>\/\/\/\/\/\/\/<\/p>\n<p><strong>Functions of several real variables<\/strong>:<\/p>\n<p>A function <span class=\"wp-katex-eq\" data-display=\"false\"> f <\/span> from <span class=\"wp-katex-eq\" data-display=\"false\"> \\mathbb{R}^2 <\/span> to <span class=\"wp-katex-eq\" data-display=\"false\"> \\mathbb{R} <\/span> can be described by:<\/p>\n<p><span class=\"wp-katex-eq\" data-display=\"false\"> {{\\mathbb{R}^2 \\, \\stackrel {f} {\\longrightarrow} \\, \\mathbb{R} \\:}\\atop {\\: (x,y) \\, \\longmapsto \\, f(x,y) } } {\\,} <\/span>.<\/p>\n<p>The <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/differentials\/\" target=\"_blank\" rel=\"noopener\"><em> differential <\/em><\/a> <span class=\"wp-katex-eq\" data-display=\"false\"> \\, df \\, <\/span> of the function <span class=\"wp-katex-eq\" data-display=\"false\"> f <\/span> at the point <span class=\"wp-katex-eq\" data-display=\"false\"> (a,b) \\in \\mathbb{R}^2 <\/span> is given by:<\/p>\n<p><span class=\"wp-katex-eq\" data-display=\"false\"> \\, df_{(a,b)} = \\frac{\\partial f}{\\partial x}_{(a,b)} dx + \\frac{\\partial f}{\\partial y}_{(a,b)} dy <\/span>.<\/p>\n<p>The equation of the <a title=\"Level set (at Wikipedia)\" href=\"https:\/\/en.wikipedia.org\/wiki\/Level_set\" target=\"_blank\" rel=\"noopener\"><em>level curve<\/em><\/a> (<span class=\"wp-katex-eq\" data-display=\"false\"> \\, f = \\text{constant} \\, <\/span>) of the function <span class=\"wp-katex-eq\" data-display=\"false\"> \\, f \\, <\/span> at the point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, (a,b) \\, <\/span><br \/>\nis given by:<\/p>\n<p><span class=\"wp-katex-eq\" data-display=\"false\"> \\, f(x,y)=f(a,b) <\/span>.<\/p>\n<p>The equation of the <a title=\"Tangential and normal components of a vector (at Wikipedia)\" href=\"https:\/\/en.wikipedia.org\/wiki\/Tangential_and_normal_components\" target=\"_blank\" rel=\"noopener\"><em>tangent to the level curve<\/em><\/a> of the function <span class=\"wp-katex-eq\" data-display=\"false\"> \\, f \\, <\/span> at the point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, (a,b) \\, <\/span> is given by:<\/p>\n<p><span class=\"wp-katex-eq\" data-display=\"false\"> \\, \\frac{\\partial f}{\\partial x}_{(a,b)} (x-a) + \\frac{\\partial f}{\\partial y}_{(a,b)} (y-b) = 0 <\/span>.<\/p>\n<p>The normal to this tangent at the point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, (a, b) \\, <\/span> is called the <a title=\"at Wikipedia\" href=\"https:\/\/en.wikipedia.org\/wiki\/Gradient\" target=\"_blank\" rel=\"noopener\"><em>gradient<\/em><\/a> of the function <span class=\"wp-katex-eq\" data-display=\"false\"> \\, f \\, <\/span> at the point <span class=\"wp-katex-eq\" data-display=\"false\"> \\, (a,b) \\, <\/span>. It is given by the vector<\/p>\n<p><span class=\"wp-katex-eq\" data-display=\"false\"> \\, (\\frac{\\partial f}{\\partial x}_{(a,b)}, \\frac{\\partial f}{\\partial y}_{(a,b)})<\/span>.<\/p>\n<p><strong>NOTE<\/strong>: In three dimensions the level curve is a level surface, and the tangent line is a tangent plane. The gradient is still perpendicular to the tangent plane at the point of tangency. <a title=\"at the KMR web site\" href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/gradients\/\" target=\"_blank\" rel=\"noopener\">The relation between the gradients in two and three dimensions is visualized here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This page is a sub-page of the page on Calculus. \/\/\/\/\/\/\/ The sub-pages of this page are: \u2022 Functions of Several Real Variables \u2022 Partial and Directional Derivatives \u2022 Differentiation and Affine Approximation \u2022 Gradients \u2022 The Chain Rule in Several Real Variables \u2022 Taylor Expansion in in Several Real Variables \u2022 Riemann Integration in &hellip; <a href=\"https:\/\/kmr.placify.me\/wp\/research\/math-rehab\/learning-object-repository\/calculus\/calculus-of-several-real-variables\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Calculus of  Several Real Variables<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":531,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":["post-535","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/535","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/comments?post=535"}],"version-history":[{"count":3,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/535\/revisions"}],"predecessor-version":[{"id":36704,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/535\/revisions\/36704"}],"up":[{"embeddable":true,"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/pages\/531"}],"wp:attachment":[{"href":"https:\/\/kmr.placify.me\/wp\/wp-json\/wp\/v2\/media?parent=535"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}