summaryrefslogtreecommitdiff
path: root/project/templates/integral.html
diff options
context:
space:
mode:
Diffstat (limited to 'project/templates/integral.html')
-rw-r--r--project/templates/integral.html241
1 files changed, 0 insertions, 241 deletions
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>&#x222b;</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>&#x222b;</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>&#x222b;</mo>
- <mn>0</mn>
- <mn>1</mn>
- </munderover>
- <mi>sin</mi>
- <mrow>
- <mo>(</mo>
- <mi>&#x03c0;</mi>
- <mo>&#x2062;</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>&#x03c0;</mi>
- </mfrac>
- <msubsup>
- <menclose notation="right">
- <mrow>
- <mi>cos</mi>
- <mrow>
- <mo>(</mo>
- <mi>&#x03c0;</mi>
- <mo>&#x2062;</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>&#x03c0;</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>&#x03c0;</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 -%}
-
-