{"id":570,"date":"2011-08-04T10:23:11","date_gmt":"2011-08-04T09:23:11","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=570"},"modified":"2011-08-04T10:23:11","modified_gmt":"2011-08-04T09:23:11","slug":"the-fundamental-misunderstanding-of-calculus","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2011\/08\/04\/the-fundamental-misunderstanding-of-calculus\/","title":{"rendered":"The fundamental misunderstanding of calculus"},"content":{"rendered":"

We all know the fundamental theorems of calculus, if not check Wikipedia<\/a>.\u00a0 I now want to\u00a0 demonstrate what has been called the fundamental misunderstanding of calculus<\/em>.<\/p>\n

Let us consider the two dimensional plane and equip it with coordinates \\((x,y)\\).\u00a0 Associated with this choice of coordinates are\u00a0 the partial derivatives<\/p>\n

\\(\\left( \\frac{\\partial}{\\partial x} , \\frac{\\partial}{\\partial y} \\right)\\).<\/p>\n

You can think about these in terms of the tangent sheaf etc. if so desired, but we will keep things quite simple.<\/p>\n

Now let us consider a change of coordinates. We will be quite specific here for illustration purposes<\/p>\n

\\(x \\rightarrow \\bar{x} = x +y\\),<\/p>\n

\\(y \\rightarrow \\bar{y} = y\\).<\/p>\n

Now think about how these effect the partial derivatives. This is really just a simple change of variables.\u00a0 Let me now state\u00a0 the fundamental misunderstanding of\u00a0 calculus in a way suited to our example:<\/p>\n

Misunderstanding: <\/strong>Despite <\/em>coordinate x changing the partial derivative with respect to x remains unchanged. Despite the coordinate y remaining unchanged the partial derivative with respect to y changes.<\/em><\/p>\n

This may seem at first counter intuitive, but is correct. Let us prove it.<\/p>\n

Note hat we can invert the change of coordinate for x very simply<\/p>\n

\\(x = \\bar{x} {-}\\bar{y} \\),<\/p>\n

using the fact that y does not change. Then one needs to use the chain rule<\/a>,<\/p>\n

\\(\\frac{\\partial}{\\partial \\bar{x}}\u00a0 = \\frac{\\partial x}{\\partial \\bar{x}}\\frac{\\partial}{\\partial x}+ \\frac{\\partial y}{\\partial \\bar{x}}\\frac{\\partial}{\\partial y}\u00a0\u00a0 =\u00a0\u00a0\u00a0 \\frac{\\partial}{\\partial x}\\),<\/p>\n

\\(\\frac{\\partial}{\\partial \\bar{y}}\u00a0 = \\frac{\\partial x}{\\partial \\bar{y}}\\frac{\\partial}{\\partial x}+ \\frac{\\partial y}{\\partial \\bar{y}}\\frac{\\partial}{\\partial y}\u00a0\u00a0 =\u00a0\u00a0\u00a0 \\frac{\\partial}{\\partial y} {-} \\frac{\\partial}{\\partial x} \\).<\/p>\n

There we are. Despite our initial gut feeling that that the partial derivative wrt y should remain unchanged we see that it is in fact the partial derivative wrt x that is unchanged.\u00a0 This can course some confusion the first time you see it,\u00a0 and hence the nomenclature the fundamental misunderstanding of calculus<\/em>.<\/p>\n

I apologise for forgetting who first named the misunderstanding.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

We all know the fundamental theorems of calculus, if not check Wikipedia.\u00a0 I now want to\u00a0 demonstrate what has been called the fundamental misunderstanding of calculus. Let us consider the two dimensional plane and equip it with coordinates \\((x,y)\\).\u00a0 Associated with this choice of coordinates are\u00a0 the partial derivatives \\(\\left( \\frac{\\partial}{\\partial x} , \\frac{\\partial}{\\partial y} … Continue reading The fundamental misunderstanding of calculus<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-570","post","type-post","status-publish","format-standard","hentry","category-general-mathematics"],"_links":{"self":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/570","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/comments?post=570"}],"version-history":[{"count":0,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/570\/revisions"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=570"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=570"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=570"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}