From e59ca4a3eaa53d66fb2dcd3ddbdd86d99b04b7c8 Mon Sep 17 00:00:00 2001 From: Jed Barber Date: Mon, 28 Jun 2021 00:21:26 +1200 Subject: Converted everything to XHTML 1.1 --- project/templates/integral.html | 173 ++++++++++++++++++++-------------------- 1 file changed, 87 insertions(+), 86 deletions(-) (limited to 'project/templates/integral.html') 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 %} - + {% endblock %} @@ -21,25 +21,26 @@
29/12/2018
-

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.

+

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.

Consider the following example:

- - ∫ + + ∫ 0 1 - + - + + ( -1 - + ) + x dx @@ -47,70 +48,68 @@ for complex valued functions.
-

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.

+

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.

-
+
Real values only plot of minus one raised to the x power -
Figure 1 - Real values only
-
- -

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 -Phantom Graphs. -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.

+ width="520" /> +
Figure 1 - Real values only
+ + +

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 +Phantom Graphs. 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.

Upon doing so, the following helical graph is obtained:

-
+
Complete three dimensional graph of all values of minux one raised to the x power -
Figure 2 - Full graph
-
+ width="520" /> +
Figure 2 - Full graph
+ -

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.

+

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.

-

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.

+

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.

-
+
Graph of the real component of minus one raised to the power of x -
Figure 3 - Real component
-
+ width="520" /> +
Figure 3 - Real component
+ 		-
+
Graph of the imaginary component of minus one raised to the power of x -
Figure 4 - Imaginary component
-
+ width="520" /> +
Figure 4 - Imaginary component
+
-

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:

+

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:

@@ -118,15 +117,17 @@ consideration, the real values disappear and we are left with the following:

@@ -217,9 +218,9 @@ consideration, the real values disappear and we are left with the following:

2 - i + i - π + π @@ -228,14 +229,14 @@ consideration, the real values disappear and we are left with the following:

- - ∫ + + ∫ 0 1 - + - + + ( -1 - + ) + x dx @@ -140,19 +141,19 @@ consideration, the real values disappear and we are left with the following:

i - - ∫ + + ∫ 0 1 - + sin - - - π - ⁢ - x - - + + ( + π + ⁢ + x + ) + dx @@ -166,19 +167,19 @@ consideration, the real values disappear and we are left with the following:

- i - π + π cos - - - π - ⁢ - x - - + + ( + π + ⁢ + x + ) + 0 @@ -196,15 +197,15 @@ consideration, the real values disappear and we are left with the following:

- i - π + π - - - -1 - - - 1 - - + + ( + -1 + - + 1 + ) +
-

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.

+

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.

The next interesting question is how this relates to the surface area of a -right helicoid.

+ +right helicoid.

{% endblock %} -- cgit