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diff --git a/project/templates/integral.html b/project/templates/integral.html deleted file mode 100644 index 364fe84..0000000 --- a/project/templates/integral.html +++ /dev/null @@ -1,241 +0,0 @@ - -{%- extends "base_math.html" -%} - - - -{%- block title -%}Area Under the Curve of a Complex Integral{%- endblock -%} - - - -{%- block style %} - <link href="/css/integral.css" rel="stylesheet" /> -{% endblock -%} - - - -{%- block content %} -<h4>Area Under the Curve of a Complex Integral</h4> - -<p>Scripts used to generate graphs: <a href="/dld/integral_scripts.zip">Link</a></p> - -<h5>29/12/2018</h5> - -<p>A definite integral can be represented on the xy-plane as the signed area bounded by the curve of -the function f(x), the x-axis, and the limits of integration a and b. But it's not immediately clear -how this definition applies for complex valued functions.</p> - -<p>Consider the following example:</p> - -<div class="precontain"><div class="mathblock"> - <math xmlns="http://www.w3.org/1998/Math/MathML"> - <mrow> - <munderover> - <mo>∫</mo> - <mn>0</mn> - <mn>1</mn> - </munderover> - <msup> - <mrow> - <mo>(</mo> - <mn>-1</mn> - <mo>)</mo> - </mrow> - <mi>x</mi> - </msup> - <mi>dx</mi> - </mrow> - </math> -</div></div> - -<p>If the function is graphed on the xy-plane, the real valued outputs are sparse. Yet an elementary -antiderivative exists and the definite integral is well defined.</p> - -<div class="figure"> - <img src="/img/minus_one_exp_x_real_values_only.png" - alt="Real values only plot of minus one raised to the x power" - height="400" - width="520" /> - <div class="figcaption">Figure 1 - Real values only</div> -</div> - -<p>In order to plot a meaningful graph that can be used to potentially calculate the integral as a -signed area, some cues are taken from Philip Lloyd's work on -<a href="https://phantomgraphs.weebly.com/" class="external">Phantom Graphs</a>. In that work, an -additional z-axis is used to extend the x-axis into a complex xz-plane, allowing complex inputs to -be graphed. For the function considered here, the z-axis is instead used to extend the y-axis into a -complex yz-plane to allow graphing of complex outputs instead.</p> - -<p>Upon doing so, the following helical graph is obtained:</p> - -<div class="figure"> - <img src="/img/minus_one_exp_x_full_plot.png" - alt="Complete three dimensional graph of all values of minux one raised to the x power" - height="400" - width="520" /> - <div class="figcaption">Figure 2 - Full graph</div> -</div> - -<p>The curve is continuous and spirals around the x-axis, intersecting with the real xy-plane at the -points plotted in the initial graph of Figure 1. However it is still not clear how to represent the -area under the curve.</p> - -<p>Observing that complex numbers in cartesian form are composed of a real part and an imaginary -part, it is possible to decompose the function into real and imaginary components. These are easy to -obtain by rotating the graph above to view the real and imaginary parts as flat planes.</p> - -<table id="component"> - <tr> - <td> - <div class="figure"> - <img src="/img/cos_pi_x.png" - alt="Graph of the real component of minus one raised to the power of x" - height="400" - width="520" /> - <div class="figcaption">Figure 3 - Real component</div> - </div> - </td> - <td> - <div class="figure"> - <img src="/img/sin_pi_x.png" - alt="Graph of the imaginary component of minus one raised to the power of x" - height="400" - width="520" /> - <div class="figcaption">Figure 4 - Imaginary component</div> - </div> - </td> - </tr> -</table> - -<p>From this it can be seen that the function is a combination of a real valued cosine and an -imaginary valued sine. With the limits of integration under consideration, the real values disappear -and we are left with the following:</p> - -<div class="precontain"><div class="mathblock"> - <table> - <tr> - <td colspan="2"> - <math xmlns="http://www.w3.org/1998/Math/MathML"> - <mrow> - <munderover> - <mo>∫</mo> - <mn>0</mn> - <mn>1</mn> - </munderover> - <msup> - <mrow> - <mo>(</mo> - <mn>-1</mn> - <mo>)</mo> - </mrow> - <mi>x</mi> - </msup> - <mi>dx</mi> - </mrow> - </math> - </td> - </tr> - <tr> - <td>=</td> - <td> - <math xmlns="http://www.w3.org/1998/Math/MathML"> - <mrow> - <mi>i</mi> - <munderover> - <mo>∫</mo> - <mn>0</mn> - <mn>1</mn> - </munderover> - <mi>sin</mi> - <mrow> - <mo>(</mo> - <mi>π</mi> - <mo>⁢</mo> - <mi>x</mi> - <mo>)</mo> - </mrow> - <mi>dx</mi> - </mrow> - </math> - </td> - </tr> - <tr> - <td>=</td> - <td> - <math xmlns="http://www.w3.org/1998/Math/MathML"> - <mrow> - <mo>-</mo> - <mfrac> - <mi>i</mi> - <mi>π</mi> - </mfrac> - <msubsup> - <menclose notation="right"> - <mrow> - <mi>cos</mi> - <mrow> - <mo>(</mo> - <mi>π</mi> - <mo>⁢</mo> - <mi>x</mi> - <mo>)</mo> - </mrow> - </mrow> - </menclose> - <mn>0</mn> - <mn>1</mn> - </msubsup> - </mrow> - </math> - </td> - </tr> - <tr> - <td>=</td> - <td> - <math xmlns="http://www.w3.org/1998/Math/MathML"> - <mrow> - <mo>-</mo> - <mfrac> - <mi>i</mi> - <mi>π</mi> - </mfrac> - <mrow> - <mo>(</mo> - <mn>-1</mn> - <mo>-</mo> - <mn>1</mn> - <mo>)</mo> - </mrow> - </mrow> - </math> - </td> - </tr> - <tr> - <td>=</td> - <td> - <math xmlns="http://www.w3.org/1998/Math/MathML"> - <mrow> - <mfrac> - <mrow> - <mn>2</mn> - <mi>i</mi> - </mrow> - <mi>π</mi> - </mfrac> - </mrow> - </math> - </td> - </tr> - </table> -</div></div> - -<p>This agrees with the answer obtained by ordinary evaluation of the integral without considering -the graph, so the informal area under the curve definition still works. Considering the area under -the curve using polar coordinates also works, but requires evaluating a less than pleasant infinite -sum and so won't be considered here.</p> - -<p>The next interesting question is how this relates to the surface area of a -<a href="https://www.mathcurve.com/surfaces.gb/helicoiddroit/helicoiddroit.shtml" class="external"> -right helicoid</a>.</p> -{% endblock -%} - - |