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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>∫</mo> + <munderover> + <mo>∫</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>∫</mo> + <munderover> + <mo>∫</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>∫</mo> + <munderover> + <mo>∫</mo> <mn>0</mn> <mn>1</mn> - </msubsup> + </munderover> <mi>sin</mi> - <mfenced> - <mrow> - <mi>π</mi> - <mo>⁢</mo> - <mi>x</mi> - </mrow> - </mfenced> + <mrow> + <mo>(</mo> + <mi>π</mi> + <mo>⁢</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>π</mi> + <mi>π</mi> </mfrac> <msubsup> <menclose notation="right"> <mrow> <mi>cos</mi> - <mfenced> - <mrow> - <mi>π</mi> - <mo>⁢</mo> - <mi>x</mi> - </mrow> - </mfenced> + <mrow> + <mo>(</mo> + <mi>π</mi> + <mo>⁢</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>π</mi> + <mi>π</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>π</mi> + <mi>π</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 %} |