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authorJed Barber <jedb@bootes.lan>2021-06-28 00:21:26 +1200
committerJed Barber <jedb@bootes.lan>2021-06-28 00:21:26 +1200
commite59ca4a3eaa53d66fb2dcd3ddbdd86d99b04b7c8 (patch)
treeba3223acbe2e99513adc0ecd9812f188a1a4ad2d /project/templates/integral.html
parent3a1a7a4d62dce554436aa8c23e980204a481c7b4 (diff)
Converted everything to XHTML 1.1
Diffstat (limited to 'project/templates/integral.html')
-rw-r--r--project/templates/integral.html173
1 files changed, 87 insertions, 86 deletions
diff --git a/project/templates/integral.html b/project/templates/integral.html
index 0e5307b..1c9d175 100644
--- a/project/templates/integral.html
+++ b/project/templates/integral.html
@@ -1,5 +1,5 @@
-{% extends "base.html" %}
+{% extends "base_math.html" %}
@@ -8,7 +8,7 @@
{% block style %}
- <link href="/css/integral.css" rel="stylesheet">
+ <link href="/css/integral.css" rel="stylesheet" />
{% endblock %}
@@ -21,25 +21,26 @@
<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>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>
- <msubsup>
- <mo>&int;</mo>
+ <munderover>
+ <mo>&#x222b;</mo>
<mn>0</mn>
<mn>1</mn>
- </msubsup>
+ </munderover>
<msup>
- <mfenced>
+ <mrow>
+ <mo>(</mo>
<mn>-1</mn>
- </mfenced>
+ <mo>)</mo>
+ </mrow>
<mi>x</mi>
</msup>
<mi>dx</mi>
@@ -47,70 +48,68 @@ for complex valued functions.</p>
</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>
+<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>
-<figure>
+<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">
- <figcaption>Figure 1 - Real values only</figcaption>
-</figure>
-
-<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/" target="_blank">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>
+ 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>
-<figure>
+<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">
- <figcaption>Figure 2 - Full graph</figcaption>
-</figure>
+ 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>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>
+<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>
- <figure>
+ <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">
- <figcaption>Figure 3 - Real component</figcaption>
- </figure>
+ width="520" />
+ <div class="figcaption">Figure 3 - Real component</div>
+ </div>
</td>
<td>
- <figure>
+ <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">
- <figcaption>Figure 4 - Imaginary component</figcaption>
- </figure>
+ 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>
+<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>
@@ -118,15 +117,17 @@ consideration, the real values disappear and we are left with the following:</p>
<td colspan="2">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
- <msubsup>
- <mo>&int;</mo>
+ <munderover>
+ <mo>&#x222b;</mo>
<mn>0</mn>
<mn>1</mn>
- </msubsup>
+ </munderover>
<msup>
- <mfenced>
+ <mrow>
+ <mo>(</mo>
<mn>-1</mn>
- </mfenced>
+ <mo>)</mo>
+ </mrow>
<mi>x</mi>
</msup>
<mi>dx</mi>
@@ -140,19 +141,19 @@ consideration, the real values disappear and we are left with the following:</p>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
<mi>i</mi>
- <msubsup>
- <mo>&int;</mo>
+ <munderover>
+ <mo>&#x222b;</mo>
<mn>0</mn>
<mn>1</mn>
- </msubsup>
+ </munderover>
<mi>sin</mi>
- <mfenced>
- <mrow>
- <mi>&pi;</mi>
- <mo>&it;</mo>
- <mi>x</mi>
- </mrow>
- </mfenced>
+ <mrow>
+ <mo>(</mo>
+ <mi>&#x03c0;</mi>
+ <mo>&#x2062;</mo>
+ <mi>x</mi>
+ <mo>)</mo>
+ </mrow>
<mi>dx</mi>
</mrow>
</math>
@@ -166,19 +167,19 @@ consideration, the real values disappear and we are left with the following:</p>
<mo>-</mo>
<mfrac>
<mi>i</mi>
- <mi>&pi;</mi>
+ <mi>&#x03c0;</mi>
</mfrac>
<msubsup>
<menclose notation="right">
<mrow>
<mi>cos</mi>
- <mfenced>
- <mrow>
- <mi>&pi;</mi>
- <mo>&it;</mo>
- <mi>x</mi>
- </mrow>
- </mfenced>
+ <mrow>
+ <mo>(</mo>
+ <mi>&#x03c0;</mi>
+ <mo>&#x2062;</mo>
+ <mi>x</mi>
+ <mo>)</mo>
+ </mrow>
</mrow>
</menclose>
<mn>0</mn>
@@ -196,15 +197,15 @@ consideration, the real values disappear and we are left with the following:</p>
<mo>-</mo>
<mfrac>
<mi>i</mi>
- <mi>&pi;</mi>
+ <mi>&#x03c0;</mi>
</mfrac>
- <mfenced>
- <mrow>
- <mn>-1</mn>
- <mo>-</mo>
- <mn>1</mn>
- </mrow>
- </mfenced>
+ <mrow>
+ <mo>(</mo>
+ <mn>-1</mn>
+ <mo>-</mo>
+ <mn>1</mn>
+ <mo>)</mo>
+ </mrow>
</mrow>
</math>
</td>
@@ -217,9 +218,9 @@ consideration, the real values disappear and we are left with the following:</p>
<mfrac>
<mrow>
<mn>2</mn>
- <mi>i</mn>
+ <mi>i</mi>
</mrow>
- <mi>&pi;</mi>
+ <mi>&#x03c0;</mi>
</mfrac>
</mrow>
</math>
@@ -228,14 +229,14 @@ consideration, the real values disappear and we are left with the following:</p>
</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>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" target="_blank">right helicoid</a>.</p>
+<a href="https://www.mathcurve.com/surfaces.gb/helicoiddroit/helicoiddroit.shtml" class="external">
+right helicoid</a>.</p>
{% endblock %}