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Edits to Chapters 13, 14, and 15 #451

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360b3d9
Edits to chapter 13 and 14
APEXCalculus Aug 14, 2026
00b6f58
Final edits to chapters 13, 14 and 15
APEXCalculus Aug 14, 2026
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8 changes: 4 additions & 4 deletions ptx/sec_center_of_mass.ptx
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Expand Up @@ -177,7 +177,7 @@
That is, for a small enough subregion of <m>R</m>,
the density across that region is almost constant.
</p>
<aside vshift="2">
<aside vshift="3">
<p>
<em>Mass</em> and <em>weight</em> are different measures.
Since they are scalar multiples of each other,
Expand Down Expand Up @@ -230,7 +230,7 @@
As the density is constant, it does not matter where we place the square.
</p>

<figure xml:id="fig_mass1" vshift="1">
<figure xml:id="fig_mass1" vshift="0">
<caption>A region <m>R</m> representing a lamina in <xref ref="ex_mass1"/></caption>
<!-- START figures/fig_mass1.tex -->
<image width="47%">
Expand Down Expand Up @@ -292,7 +292,7 @@
(see <xref ref="fig_mass1"/>),
with variable density <m>\delta(x,y) = (x+y+2)\,\text{g/cm}^2</m>.
</p>
<figure xml:id="fig_mass2" vshift="-2">
<figure xml:id="fig_mass2" vshift="-3">
<caption>Graphing the density functions in <xref ref="ex_mass1"/> and <xref ref="ex_mass2"/></caption>

<!-- START figures/figmass2_3D.asy -->
Expand Down Expand Up @@ -977,7 +977,7 @@
</p>
</sidebyside>

<figure xml:id="fig_mass5" vshift="0">
<figure xml:id="fig_mass5" vshift="2">
<caption>Illustrating the center of mass of a discrete planar system in <xref ref="ex_mass5"/></caption>
<!-- START figures/fig_mass5.tex -->
<image width="47%">
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152 changes: 77 additions & 75 deletions ptx/sec_cylindrical_spherical.ptx
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Expand Up @@ -9,20 +9,21 @@
coordinates give us new ways of desribing surfaces and regions in space.
</p>

<aside vshift="-1">
<p>
Both coordinate systems provide ways of extending polar coordinates in the plane to three dimensions.
The reader is warned that while conventions are fairly standard for cylindrical coordinates,
there are many different conventions for spherical coordinates.
Mathematics, physics, and engineering all use slightly different versions.
The definition presented in this text also differs from the usual mathematical definition.
Why add yet another competing standard?
The definition we present is slightly easier to work with,
and should also be more familiar,
since it corresponds to the coordinate system of latitude and longitude used to describe locations on the Earth.
</p>
</aside>
<p>
Both coordinate systems provide ways of extending polar coordinates in the plane to three dimensions.
The reader is warned that while conventions are fairly standard for cylindrical coordinates,
there are many different conventions for spherical coordinates, with
mathematics, physics, and engineering each using slightly different versions.
Additionally, the definition presented in this text also differs from the usual mathematical definition.
This non-standard approach is used as it is more intuitive, corresponding to the coordinate system of
latitude and longitude used to describe locations on the Earth.
</p>

<p>
Once one understands the principles of spherical coordinates, it is not difficult to translate those principles
to align with the conventions used in other disciplines.
</p>

<!-- <figure xml:id="vid-multint-cylindrical-intro" component="video">
<caption>A triple integral that isn't easily performed in rectancular coordinates</caption>
<video youtube="Ajd8uy3U4lA" label="vid-multint-cylindrical-intro"/>
Expand Down Expand Up @@ -236,7 +237,7 @@
This plane is the same as the plane described by <m>z=2</m> in rectangular coordinates.
</p>

<figure xml:id="fig_cylindrical1" vshift="0">
<figure xml:id="fig_cylindrical1" vshift="2">
<caption>Graphing the canonical surfaces in cylindrical coordinates from <xref ref="ex_cylindrical1"/></caption>
<image width="47%">
<shortdescription>Three surfaces in space, corresponding to fixed values of each of the three cylindrical coordinates.</shortdescription>
Expand Down Expand Up @@ -443,7 +444,7 @@
using a triple integral in cylindrical coordinates.
Distances are measured in centimeters and density is measured in grams per cubic centimeter.
</p>
<figure xml:id="fig_cylindrical2" vshift="-4">
<figure xml:id="fig_cylindrical2" vshift="-2">
<caption>Visualizing the solid used in <xref ref="ex_cylindrical2"/></caption>
<image width="47%">
<shortdescription>A circular cylinder capped by a spherical dome.</shortdescription>
Expand Down Expand Up @@ -728,70 +729,15 @@
<m>0 \leq \theta \leq 2\pi</m> and <m>-\pi/2 \leq \varphi \leq \pi/2</m>.
</p>

<aside vshift="5">
<aside vshift="-4">
<p>
The symbol <m>\rho</m> is the Greek letter <q>rho.</q>
Traditionally it is used in the spherical coordinate system,
while <m>r</m> is used in the polar and cylindrical coordinate systems.
</p>
</aside>

<convention xml:id="convention-spherical">
<p>
Note that most mathematics textbooks define <m>\varphi</m> to be measured
from the positive <m>z</m>-axis, with values in <m>[0,\pi]</m>,
rather than from the <m>xy</m>-plane.
</p>

<p>
We have chosen our convention with a number of considerations in mind:
<ul>
<li>
<p>
The coordinates <m>(\rho,\theta,\varphi)</m> form a <term>right-handed</term>
coordinate system: one in which the orientation matches that of our usual <m>(x,y,z)</m>
coordinates, where the <q>right-hand rule</q> applies.
If <m>\varphi</m> is measured from the <m>z</m>-axis,
the order <m>(\rho, \varphi, \theta)</m> is needed to get a right-handed system.
</p>
</li>
<li>
<p>
Points of the form <m>(a,\alpha,0)</m> are the same in both cylindrical and spherical coordinates.
</p>
</li>
<li>
<p>
Some integration problems become slightly easier:
we will see soon that the volume element in spherical coordinates involves <m>\cos(\varphi)</m>,
which integrates to <m>\sin(\varphi)</m>.
In the usual convention, the volume element involves <m>\sin(\varphi)</m>,
which integrates to <m>-\cos(\varphi)</m> <ndash/> a source of many common sign errors.
</p>
</li>
</ul>
</p>

<p>
Students of Physics will encounter yet another convention.
In Physics, the variable <m>r</m> is preferred as the radial coordinate,
and spherical coordinates are given as <m>(r,\theta,\varphi)</m>;
however, in Physics, <m>\varphi</m> becomes the angle in the <m>xy</m>-plane,
while <m>\theta</m> is the angle measured from the positive <m>z</m>-axis.
</p>

<p>
Note that the angle in the <m>xy</m>-plane (<m>\theta</m>, in our case)
is known as the <term>azimuthal angle</term>.
Our angle <m>\varphi</m> is known as the <term>elevation angle</term>.
The angle used in other conventions that is measured from the positive <m>z</m>-axis
(often identified with the north pole) is known as the <term>polar angle</term>.
For further discussion, the <url href="https://en.wikipedia.org/wiki/Spherical_coordinate_system" visual="en.wikipedia.org/wiki/Spherical_coordinate_system">Wikipedia article</url>
is quite useful.
</p>
</convention>

<figure xml:id="fig_sphericalintro" vshift="11">
<figure xml:id="fig_sphericalintro" vshift="2">
<caption>Illustrating the principles behind spherical coordinates</caption>
<image width="47%">
<shortdescription>A schematic diagram illustrating the spherical coordinate system relative to the rectangular coordinate axes.</shortdescription>
Expand Down Expand Up @@ -909,6 +855,62 @@
</image>
</figure>

<convention xml:id="convention-spherical">
<p>
Note that most mathematics textbooks define <m>\varphi</m> to be measured
from the positive <m>z</m>-axis, with values in <m>[0,\pi]</m>,
rather than from the <m>xy</m>-plane.
</p>

<p>
We have chosen our convention with a number of considerations in mind:
<ul>
<li>
<p>
The coordinates <m>(\rho,\theta,\varphi)</m> form a <term>right-handed</term>
coordinate system: one in which the orientation matches that of our usual <m>(x,y,z)</m>
coordinates, where the <q>right-hand rule</q> applies.
If <m>\varphi</m> is measured from the <m>z</m>-axis,
the order <m>(\rho, \varphi, \theta)</m> is needed to get a right-handed system.
</p>
</li>
<li>
<p>
Points of the form <m>(a,\alpha,0)</m> are the same in both cylindrical and spherical coordinates.
</p>
</li>
<li>
<p>
Some integration problems become slightly easier:
we will see soon that the volume element in spherical coordinates involves <m>\cos(\varphi)</m>,
which integrates to <m>\sin(\varphi)</m>.
In the usual convention, the volume element involves <m>\sin(\varphi)</m>,
which integrates to <m>-\cos(\varphi)</m> <ndash/> a source of many common sign errors.
</p>
</li>
</ul>
</p>

<p>
Students of Physics will encounter yet another convention.
In Physics, the variable <m>r</m> is preferred as the radial coordinate,
and spherical coordinates are given as <m>(r,\theta,\varphi)</m>;
however, in Physics, <m>\varphi</m> becomes the angle in the <m>xy</m>-plane,
while <m>\theta</m> is the angle measured from the positive <m>z</m>-axis.
</p>

<p>
Note that the angle in the <m>xy</m>-plane (<m>\theta</m>, in our case)
is known as the <term>azimuthal angle</term>.
Our angle <m>\varphi</m> is known as the <term>elevation angle</term>.
The angle used in other conventions that is measured from the positive <m>z</m>-axis
(often identified with the north pole) is known as the <term>polar angle</term>.
For further discussion, the <url href="https://en.wikipedia.org/wiki/Spherical_coordinate_system" visual="en.wikipedia.org/wiki/Spherical_coordinate_system">Wikipedia article</url>
is quite useful.
</p>
</convention>


<figure xml:id="vid-multint-spherical-intro" component="video" vshift="4">
<caption>Introducing spherical coordinates</caption>
<video youtube="In5up-a1jMI" label="vid-multint-spherical-intro"/>
Expand Down Expand Up @@ -1037,7 +1039,7 @@
with the positive <m>z</m>-axis its axis of symmetry, with point at the origin.
</p>

<figure xml:id="fig_spherical1" vshift="-1">
<figure xml:id="fig_spherical1" vshift="3">
<caption>Graphing the canonical surfaces in spherical coordinates from <xref ref="ex_spherical1"/></caption>
<image width="47%">
<shortdescription>A three-dimensional plot showing three surfaces, each of which is obtained by fixing a value of one of the three spherical coordinates.</shortdescription>
Expand Down Expand Up @@ -1197,7 +1199,7 @@
</md>.
</p>

<figure xml:id="fig_sphericalwedge" vshift="-3">
<figure xml:id="fig_sphericalwedge" vshift="1">
<caption>Approximating the volume of a standard region in space using spherical coordinates</caption>
<image width="47%">
<shortdescription>A schematic diagram illustrating how the spherical volume element is computed.</shortdescription>
Expand Down Expand Up @@ -1412,7 +1414,7 @@
Again, this development of <m>dV</m> should sound reasonable,
and the following theorem states it is the appropriate manner by which triple integrals are to be evaluated in spherical coordinates.
</p>
<aside vshift="-7">
<aside vshift="-2">
<p>
It is generally most intuitive to evaluate the triple integral in <xref ref="thm_triple_int_spherical"/>
by integrating with respect to <m>\rho</m> first;
Expand Down
11 changes: 6 additions & 5 deletions ptx/sec_double_int_polar.ptx
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Expand Up @@ -590,7 +590,7 @@
<m>\ds f(x,y) =\frac1{x^2+y^2+1}</m> over the sector of the circle with radius <m>a</m> centered at the origin in the first quadrant,
as shown in <xref ref="fig_doublepol5"/>.
</p>
<figure xml:id="fig_doublepol5">
<figure xml:id="fig_doublepol5" vshift="0">
<caption>The surface and region <m>R</m> used in <xref ref="ex_doublepol5"/></caption>

<!-- START figures/figdoublepol5_3D.asy -->
Expand Down Expand Up @@ -619,7 +619,7 @@

size(200,200,IgnoreAspect);
//currentprojection=perspective(7,2,1);
currentprojection=orthographic(8,-13,2.8);
currentprojection=orthographic(13,3,3);
defaultrender.merge=true;

// setup and draw the axes
Expand Down Expand Up @@ -696,7 +696,7 @@
<mrow>\amp = \frac{\pi}{4}\ln(a^2+1)</mrow>
</md>.
</p>
<aside vshift="0">
<aside vshift="2">
<p>
Previous work has shown that there is finite <em>area</em>
under <m>\frac{1}{x^2+1}</m> over the entire <m>x</m>-axis.
Expand Down Expand Up @@ -981,8 +981,9 @@
</pg-code> -->
<statement>
<p>
Match the correct elements on the left to the corresponding elements on the right,
so that <m>\iint_R f(x,y)\, dA</m> is correctly converted to polar coordinates.
To convert <m>\iint_R f(x,y)\, dA</m> to polar coordinates, each of <m>x</m>, <m>y</m>, and <m>dA</m>
need to be replaced with the proper elements. For each element in the list at left,
choose the correct element from the list at right.
</p>
</statement>
<cardsort>
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2 changes: 1 addition & 1 deletion ptx/sec_double_int_volume.ptx
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Expand Up @@ -337,7 +337,7 @@
This leads to a definition.
</p>

<aside xml:id="note_doubleint" vshift="-2">
<aside xml:id="note_doubleint" vshift="-4">
<title>Double integrals as limits of double sums</title>

<p>
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8 changes: 4 additions & 4 deletions ptx/sec_greensthm.ptx
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Expand Up @@ -699,7 +699,7 @@
find the circulation of <m>\vec F</m> around the boundary of <m>R</m> and show that is equal to the double integral of <m>\curl \vec F</m> over <m>R</m>.
</p>

<figure xml:id="fig_green1" vshift="0">
<figure xml:id="fig_green1" vshift="-2">
<caption>The vector field and planar region used in <xref ref="ex_green1"/></caption>
<image width="47%">
<shortdescription>A triangular path in the plane is plotted against a two-dimensional vector field.</shortdescription>
Expand Down Expand Up @@ -875,7 +875,7 @@
Find the circulation around <m>C</m>.
</p>

<figure xml:id="fig_green2" vshift="0">
<figure xml:id="fig_green2" vshift="-2">
<caption>The vector field and planar region used in <xref ref="ex_green2"/></caption>
<image width="47%">
<shortdescription>A curve in the plane, like a bumpy circle or flower, and a vector field.</shortdescription>
Expand Down Expand Up @@ -1178,7 +1178,7 @@
that is, find the flux across <m>C</m> and show it is equal to the double integral of <m>\divv \vec F</m> over <m>R</m>.
</p>

<figure xml:id="fig_div1" vshift="-5.5">
<figure xml:id="fig_div1" vshift="1">
<caption>The region <m>R</m> used in <xref ref="ex_div1"/></caption>
<image width="47%">
<shortdescription>A circle in the plane, plotted against a spiral vector field.</shortdescription>
Expand Down Expand Up @@ -1300,7 +1300,7 @@
consider:
</p>

<figure xml:id="fig_div2" vshift="-1">
<figure xml:id="fig_div2" vshift="-3">
<caption>As used in <xref ref="ex_div2"/>, the vector field has a divergence of 0 and the two paths only intersect at their initial and terminal points.</caption>
<image width="47%">
<shortdescription>Two curves in the plane between points A and B, and a vector field with many vortices.</shortdescription>
Expand Down
2 changes: 1 addition & 1 deletion ptx/sec_iterated_integrals.ptx
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Expand Up @@ -689,7 +689,7 @@
indicating that <m>R</m> is a triangle.
</p>

<figure xml:id="fig_double5" vshift="0">
<figure xml:id="fig_double5" vshift="2">
<caption>Sketching the region <m>R</m> described by the iterated integral in <xref ref="ex_double5"/></caption>
<!-- START figures/fig_double5.tex -->
<image width="47%">
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2 changes: 1 addition & 1 deletion ptx/sec_line_int_intro.ptx
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Expand Up @@ -373,7 +373,7 @@
</statement>
</definition>

<aside vshift="0">
<aside vshift="2">
<p>
<em>Note:</em> <xref ref="def_line_integral1"/>
uses the term <em>scalar field</em>
Expand Down
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