Converted everything to XHTML 1.1

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 %}