{"id":113,"date":"2012-06-21T20:09:56","date_gmt":"2012-06-22T03:09:56","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=113"},"modified":"2012-06-21T20:09:56","modified_gmt":"2012-06-22T03:09:56","slug":"interpolating-between-vectorized-greens-theorems","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/06\/interpolating-between-vectorized-greens-theorems\/","title":{"rendered":"Interpolating Between Vectorized Green's Theorems"},"content":{"rendered":"<p>Green's theorem says that (subject to some very reasonable conditions that we need not concern ourselves with here) the counterclockwise line integral of the vector field <strong>F<\/strong> = [P Q] around the boundary of a region is equal to the double intregral of <span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_1d6b62d0d560aee32d0dc66ac09129e7.gif' style='vertical-align: middle; border: none; ' class='tex' alt=\"\\frac{\\partial Q}{\\partial x}-\\frac{\\partial P}{\\partial y}\" \/><\/span><script type='math\/tex'>\\frac{\\partial Q}{\\partial x}-\\frac{\\partial P}{\\partial y}<\/script> over the region itself. It's natural to think of it as a special case of Stokes's theorem in the case of a plane. We can also think of the line integral as the integral of the inner product of the vector field with the unit tangent, leading us to write Green's theorem like this:<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_89c4a1bba29f14991dbdeb8aabf4e616.gif' style='vertical-align: middle; border: none;' class='tex' alt=\" \\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot\\vec{\\mathbf{T}}\\, ds=\\iint_{D}(\\mathrm{curl\\,}\\vec{\\mathbf{F}})\\cdot\\vec{\\mathbf{k}}\\, ds\" \/><\/span><script type='math\/tex;  mode=display'> \\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot\\vec{\\mathbf{T}}\\, ds=\\iint_{D}(\\mathrm{curl\\,}\\vec{\\mathbf{F}})\\cdot\\vec{\\mathbf{k}}\\, ds<\/script><\/p><\/p>\n<p>But some texts (I have Mardsen and Tromba's <em>Vector Calculus<\/em> and Stewart's <em>Calculus: Early Transcendentals<\/em> in my possession; undoubtedly there are others) point out that we can also think of Green's theorem as a special case of the divergence theorem! Suppose we take the integral of the inner product of the vector field with the outward-facing unit normal (instead of the unit tangent)&mdash;it turns out that<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_2eab50d13045729964401becba1b3576.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot\\vec{\\mathbf{n}}\\, ds=\\iint_{D}\\mathrm{div\\,}\\vec{\\mathbf{F}} ds\" \/><\/span><script type='math\/tex;  mode=display'>\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot\\vec{\\mathbf{n}}\\, ds=\\iint_{D}\\mathrm{div\\,}\\vec{\\mathbf{F}} ds<\/script><\/p><\/p>\n<p>&mdash;which suggests that there's some deep fundamental sense in which Stokes's theorem and the divergence theorem are really just <em>mere surface manifestations of one and the same underlying idea<\/em>! (I'm told that it's called the generalized Stokes's theorem, but regrettably I don't know the details yet.)<\/p>\n<p><!--more--><\/p>\n<p>But here's something I thought was pretty. We have these two equations, one of which involves the unit tangent vector, and one of which involves the unit normal vector. (It's actually pointing in the opposite direction of the unit normal that we use to define the Frenet frame, which one might argue is an unfortunate clash of conventions, but whatever.) So my linear-algebraic intuitions say: hey, why not smoosh them together in a linear combination? (Making linear combinations is just what you <em>do<\/em> when you see orthogonal unit vectors, right?) Then it's like we're interpolating between the two forms of Green's theorem. Specifically, let \u03b8 be a constant. Then we have<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_af5a3a1ef03eafd2fe2f9dd53461a6b6.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot((\\sin\\theta)\\vec{\\mathbf{T}}+(\\cos\\theta)\\vec{\\mathbf{n}})\\, ds\" \/><\/span><script type='math\/tex;  mode=display'>\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot((\\sin\\theta)\\vec{\\mathbf{T}}+(\\cos\\theta)\\vec{\\mathbf{n}})\\, ds<\/script><\/p> (component decomposition of some unit vector)<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_26aa3479badbbd0e08012d661661a9e2.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"=\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot(\\sin\\theta)\\vec{\\mathbf{T}}+\\vec{\\mathbf{F}}\\cdot(\\cos\\theta)\\vec{\\mathbf{n}}\\, ds\" \/><\/span><script type='math\/tex;  mode=display'>=\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot(\\sin\\theta)\\vec{\\mathbf{T}}+\\vec{\\mathbf{F}}\\cdot(\\cos\\theta)\\vec{\\mathbf{n}}\\, ds<\/script><\/p><\/p>\n<p>(inner product distributes over vector addition)<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_e05214b4927732f44823097b13d0c6a0.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"=(\\sin\\theta)\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot\\vec{\\mathbf{T}}\\, ds+(\\cos\\theta)\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot\\vec{\\mathbf{n}}\\, ds\" \/><\/span><script type='math\/tex;  mode=display'>=(\\sin\\theta)\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot\\vec{\\mathbf{T}}\\, ds+(\\cos\\theta)\\oint_{\\partial D}\\vec{\\mathbf{F}}\\cdot\\vec{\\mathbf{n}}\\, ds<\/script><\/p><\/p>\n<p>(linearity of integrals, bilinearity of inner product)<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_f647cef7373c970d41644351e1efe85c.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"=(\\sin\\theta)\\iint_{D}(\\mathrm{curl\\,}\\vec{\\mathbf{F}})\\cdot\\vec{\\mathbf{k}}\\, dA+(\\cos\\theta)\\iint_{D}\\mathrm{div\\,}\\vec{\\mathbf{F}} dA\" \/><\/span><script type='math\/tex;  mode=display'>=(\\sin\\theta)\\iint_{D}(\\mathrm{curl\\,}\\vec{\\mathbf{F}})\\cdot\\vec{\\mathbf{k}}\\, dA+(\\cos\\theta)\\iint_{D}\\mathrm{div\\,}\\vec{\\mathbf{F}} dA<\/script><\/p><\/p>\n<p>(vectorized Green's theorem)<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_c9adf99835bf43a1ebe68b760debe65f.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"=(\\sin\\theta)\\iint_{D}\\left(\\frac{\\partial Q}{\\partial x}-\\frac{\\partial P}{\\partial y}\\right)\\, dA+(\\cos\\theta)\\iint_{D}\\left(\\frac{\\partial P}{\\partial x}+\\frac{\\partial Q}{\\partial y}\\right)dA\" \/><\/span><script type='math\/tex;  mode=display'>=(\\sin\\theta)\\iint_{D}\\left(\\frac{\\partial Q}{\\partial x}-\\frac{\\partial P}{\\partial y}\\right)\\, dA+(\\cos\\theta)\\iint_{D}\\left(\\frac{\\partial P}{\\partial x}+\\frac{\\partial Q}{\\partial y}\\right)dA<\/script><\/p><\/p>\n<p>(planar curl and divergence)<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_c3ba749363dfec1e72bd744f1d42d4c8.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"=\\iint_{D}(\\cos\\theta)\\frac{\\partial P}{\\partial x}+(\\sin\\theta)\\frac{\\partial Q}{\\partial x}-(\\sin\\theta)\\frac{\\partial P}{\\partial y}+(\\cos\\theta)\\frac{\\partial Q}{\\partial y}\\, dA\" \/><\/span><script type='math\/tex;  mode=display'>=\\iint_{D}(\\cos\\theta)\\frac{\\partial P}{\\partial x}+(\\sin\\theta)\\frac{\\partial Q}{\\partial x}-(\\sin\\theta)\\frac{\\partial P}{\\partial y}+(\\cos\\theta)\\frac{\\partial Q}{\\partial y}\\, dA<\/script><\/p><\/p>\n<p>(linearity of integrals)<\/p>\n<p>Isn't that great!? Nor is it a mere coincidence that the coefficients in the final line resemble the entries of the two-by-two matrix representing a counterclockwise rotation by \u03b8. Ideally, we would then generalize this to an arbitrary number of dimensions and gain true insight into the generalized Stokes's theorem, and thus, a facet of <em>the nature of reality itself.<\/em><\/p>\n<p>But I'm not there yet.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Green's theorem says that (subject to some very reasonable conditions that we need not concern ourselves with here) the counterclockwise line integral of the vector field F = [P Q] around the boundary of a region is equal to the &hellip; <a href=\"http:\/\/zackmdavis.net\/blog\/2012\/06\/interpolating-between-vectorized-greens-theorems\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[14],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/113"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=113"}],"version-history":[{"count":25,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/113\/revisions"}],"predecessor-version":[{"id":140,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/113\/revisions\/140"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=113"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=113"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=113"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}