{"id":49238,"date":"2025-12-13T14:06:35","date_gmt":"2025-12-13T14:06:35","guid":{"rendered":"https:\/\/idea.membrane.com\/songwriter\/?p=49238"},"modified":"2025-12-13T14:06:35","modified_gmt":"2025-12-13T14:06:35","slug":"whats-the-calculus","status":"publish","type":"post","link":"https:\/\/kingarthur.com\/songwriter\/2025\/12\/13\/whats-the-calculus\/","title":{"rendered":"What&#8217;s the Calculus"},"content":{"rendered":"<p><a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus-Best-Of.mp3\">Whats-the-Calculus-Best-Of.mp3<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus-Best-Of.mp4\">Whats-the-Calculus-Best-Of.mp4<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus.mp3\">Whats-the-Calculus.mp3<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus.mp4\">Whats-the-Calculus.mp4<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus-Pt-2.mp3\">Whats-the-Calculus-Pt-2.mp3<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus-Pt-2.mp4\">Whats-the-Calculus-Pt-2.mp4<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus-Unplugged-Underground-XXVIII.mp3\">Whats-the-Calculus-Unplugged-Underground-XXVIII.mp3<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus-Unplugged-Underground-XXVIII.mp4\">Whats-the-Calculus-Unplugged-Underground-XXVIII.mp4<\/a><br \/>\n<a href=\"http:\/\/kingarthur.com\/MegaEpix-Enormous\/multimedia\/XVIII\/Whats-the-Calculus-intro.mp3\">Whats-the-Calculus-intro.mp3<\/a><\/p>\n<p>[Intro]<br \/>\nWhat is your calculus<br \/>\n(For the rest of us)<\/p>\n<p>[Verse 1]<br \/>\nWhy not put your dot<br \/>\n(&#8230; on a plot)<br \/>\nDoes the point of view<br \/>\n(&#8230; start to skew)<\/p>\n<p>[Bridge]<br \/>\nWhat is your calculus<br \/>\n(For the rest of us)<\/p>\n<p>[Chorus]<br \/>\nIs it in a straight line<br \/>\n(&#8220;Everything will be fine&#8221;)<br \/>\nOr does it swerve and curve<br \/>\n(Equaling a mayday heyday)<\/p>\n<p>[Verse 2]<br \/>\nDo your figures align<br \/>\n(&#8230; with reality)<br \/>\nThere&#8217;s no straight line<br \/>\n(&#8230; to normality)<\/p>\n<p>[Bridge]<br \/>\nWhat is your calculus<br \/>\n(For the rest of us)<\/p>\n<p>[Chorus]<br \/>\nIs it in a straight line<br \/>\n(&#8220;Everything will be fine&#8221;)<br \/>\nOr does it swerve and curve<br \/>\n(Equaling a mayday heyday)<\/p>\n<p>[Outro]<br \/>\nQuick!<br \/>\n(Watch that hockey stick)<br \/>\nGrowth<br \/>\n(Intensity, Frequency &#8212; both)<br \/>\nThis ain&#8217;t no straight line<br \/>\n(No, not at any time)<br \/>\n[Instrumental, Whistle Solo]<br \/>\nNo, not a straight line<br \/>\n(Not aligned with fine)<br \/>\nHey!<br \/>\n(It&#8217;s a mayday heyday)<\/p>\n<p>ABOUT THE SONG AND THE SCIENCE<\/p>\n<h2 data-start=\"29\" data-end=\"63\"><strong data-start=\"32\" data-end=\"63\">What is nonlinear calculus?<\/strong><\/h2>\n<p data-start=\"65\" data-end=\"249\"><strong data-start=\"65\" data-end=\"87\">Nonlinear calculus<\/strong> refers to the branch of calculus that deals with <strong data-start=\"137\" data-end=\"164\">nonlinear relationships<\/strong> \u2014 equations or systems where the output is <em data-start=\"208\" data-end=\"213\">not<\/em> directly proportional to the input.<\/p>\n<p data-start=\"251\" data-end=\"290\">Examples of nonlinear behavior include:<\/p>\n<ul data-start=\"292\" data-end=\"617\">\n<li data-start=\"292\" data-end=\"320\">\n<p data-start=\"294\" data-end=\"320\">Exponential growth\/decay<\/p>\n<\/li>\n<li data-start=\"321\" data-end=\"340\">\n<p data-start=\"323\" data-end=\"340\">Logistic curves<\/p>\n<\/li>\n<li data-start=\"341\" data-end=\"373\">\n<p data-start=\"343\" data-end=\"373\">Chaos and strange attractors<\/p>\n<\/li>\n<li data-start=\"374\" data-end=\"410\">\n<p data-start=\"376\" data-end=\"410\">Nonlinear differential equations<\/p>\n<\/li>\n<li data-start=\"411\" data-end=\"437\">\n<p data-start=\"413\" data-end=\"437\">Climate feedback loops<\/p>\n<\/li>\n<li data-start=\"438\" data-end=\"617\">\n<p data-start=\"440\" data-end=\"500\">Anything with powers, products, or functions of functions<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"619\" data-end=\"761\">In nonlinear systems, <strong data-start=\"641\" data-end=\"719\">small changes in input can produce big, disproportionate changes in output<\/strong>, or vice versa. These systems often show:<\/p>\n<ul data-start=\"763\" data-end=\"899\">\n<li data-start=\"763\" data-end=\"781\">\n<p data-start=\"765\" data-end=\"781\">feedback loops<\/p>\n<\/li>\n<li data-start=\"782\" data-end=\"800\">\n<p data-start=\"784\" data-end=\"800\">tipping points<\/p>\n<\/li>\n<li data-start=\"801\" data-end=\"816\">\n<p data-start=\"803\" data-end=\"816\">instability<\/p>\n<\/li>\n<li data-start=\"817\" data-end=\"840\">\n<p data-start=\"819\" data-end=\"840\">multiple equilibria<\/p>\n<\/li>\n<li data-start=\"841\" data-end=\"878\">\n<p data-start=\"843\" data-end=\"878\">exponential or polynomial scaling<\/p>\n<\/li>\n<li data-start=\"879\" data-end=\"899\">\n<p data-start=\"881\" data-end=\"899\">chaotic behavior<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"901\" data-end=\"1032\">This is why nonlinear calculus is central to climate science, economics, biology, engineering, and many real-world dynamic systems.<\/p>\n<hr data-start=\"1034\" data-end=\"1037\" \/>\n<h2 data-start=\"1039\" data-end=\"1072\"><strong data-start=\"1042\" data-end=\"1072\">Is all calculus nonlinear?<\/strong><\/h2>\n<p data-start=\"1074\" data-end=\"1116\"><strong data-start=\"1074\" data-end=\"1116\">No \u2014 but most of the natural world is.<\/strong><\/p>\n<p data-start=\"1118\" data-end=\"1161\">Mathematically, calculus can be applied to:<\/p>\n<h3 data-start=\"1163\" data-end=\"1209\"><strong data-start=\"1167\" data-end=\"1209\">1. Linear functions and linear systems<\/strong><\/h3>\n<p data-start=\"1210\" data-end=\"1244\">These obey strict proportionality<\/p>\n<p data-start=\"1265\" data-end=\"1325\">Derivatives and integrals behave predictably and additively.<\/p>\n<p data-start=\"1327\" data-end=\"1420\">Linear calculus is much simpler, and many early models in physics and economics relied on it.<\/p>\n<h3 data-start=\"1422\" data-end=\"1474\"><strong data-start=\"1426\" data-end=\"1474\">2. Nonlinear functions and nonlinear systems<\/strong><\/h3>\n<p data-start=\"1475\" data-end=\"1524\">Anything that isn\u2019t strictly linear is nonlinear\u00a0Most real systems \u2014 weather, population growth, climate dynamics, biological systems, markets \u2014 are fundamentally nonlinear.<\/p>\n<hr data-start=\"1715\" data-end=\"1718\" \/>\n<h2 data-start=\"1720\" data-end=\"1765\"><strong data-start=\"1723\" data-end=\"1765\">So what exactly is nonlinear calculus?<\/strong><\/h2>\n<p data-start=\"1767\" data-end=\"1805\">It\u2019s not a separate field, but rather:<\/p>\n<h3 data-start=\"1807\" data-end=\"1905\"><strong data-start=\"1811\" data-end=\"1905\">\u201cThe application of calculus to nonlinear functions and nonlinear differential equations.\u201d<\/strong><\/h3>\n<p data-start=\"1907\" data-end=\"1921\">This includes:<\/p>\n<ul data-start=\"1923\" data-end=\"2166\">\n<li data-start=\"1923\" data-end=\"1959\">\n<p data-start=\"1925\" data-end=\"1959\">Nonlinear differential equations<\/p>\n<\/li>\n<li data-start=\"1960\" data-end=\"1991\">\n<p data-start=\"1962\" data-end=\"1991\">Nonlinear dynamical systems<\/p>\n<\/li>\n<li data-start=\"1992\" data-end=\"2014\">\n<p data-start=\"1994\" data-end=\"2014\">Bifurcation theory<\/p>\n<\/li>\n<li data-start=\"2015\" data-end=\"2031\">\n<p data-start=\"2017\" data-end=\"2031\">Chaos theory<\/p>\n<\/li>\n<li data-start=\"2032\" data-end=\"2058\">\n<p data-start=\"2034\" data-end=\"2058\">Nonlinear optimization<\/p>\n<\/li>\n<li data-start=\"2059\" data-end=\"2115\">\n<p data-start=\"2061\" data-end=\"2115\">Nonlinear PDEs (Navier\u2013Stokes, climate models, etc.)<\/p>\n<\/li>\n<li data-start=\"2116\" data-end=\"2166\">\n<p data-start=\"2118\" data-end=\"2166\">Multivariate nonlinear functions and Jacobians<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"2168\" data-end=\"2360\">In practice, <strong data-start=\"2181\" data-end=\"2244\">nonlinear = complex, sensitive, coupled, and often unstable<\/strong> \u2014 which is why nonlinear calculus is the basis for modern climate modeling, turbulence, economics, ecosystems, etc.<\/p>\n<p>Growth Curve<br \/>\nThe shape that resembles a &#8220;hockey stick&#8221;\u2014a curve that starts relatively flat and then suddenly turns upward very steeply\u2014is typically referred to mathematically as an exponential curve or an exponential growth curve. In calculus, this shape is characteristic of an exponential function where the rate of growth accelerates over time. &#8220;Hockey stick&#8221; is an informal, descriptive nickname used in climate science to highlight the sudden and dramatic change observed.<\/p>\n<p><strong>From the album &#8220;<a href=\"\/songwriter\/tag\/nonlinear\/\">Nonlinear<\/a>&#8220;<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Whats-the-Calculus-Best-Of.mp3 Whats-the-Calculus-Best-Of.mp4 Whats-the-Calculus.mp3 Whats-the-Calculus.mp4 Whats-the-Calculus-Pt-2.mp3 Whats-the-Calculus-Pt-2.mp4 Whats-the-Calculus-Unplugged-Underground-XXVIII.mp3 Whats-the-Calculus-Unplugged-Underground-XXVIII.mp4 Whats-the-Calculus-intro.mp3 [Intro] What is your calculus (For the rest of us) [Verse 1] Why not put your dot (&#8230; on a plot) Does the point of view (&#8230; start to skew) [Bridge] What is your calculus (For the rest of us) [Chorus] Is it in a straight [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","footnotes":""},"categories":[3,5,9,14],"tags":[163,4190],"class_list":["post-49238","post","type-post","status-publish","format-standard","hentry","category-4d-music","category-chaos-theory","category-daniel","category-lyrics","tag-climate-change","tag-nonlinear"],"_links":{"self":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/49238","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/comments?post=49238"}],"version-history":[{"count":8,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/49238\/revisions"}],"predecessor-version":[{"id":49258,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/posts\/49238\/revisions\/49258"}],"wp:attachment":[{"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/media?parent=49238"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/categories?post=49238"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kingarthur.com\/songwriter\/wp-json\/wp\/v2\/tags?post=49238"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}