From 360b3d90d1573b396a578a84e5c29a025a33f590 Mon Sep 17 00:00:00 2001
From: Gregory Hartman <apexcalculus@gmail.com>
Date: Fri, 14 Aug 2026 15:39:58 +0000
Subject: [PATCH 1/2] Edits to chapter 13 and 14

---
 ptx/sec_center_of_mass.ptx        |   6 +-
 ptx/sec_cylindrical_spherical.ptx | 152 +++++++++++++++---------------
 ptx/sec_double_int_polar.ptx      |  11 ++-
 ptx/sec_double_int_volume.ptx     |   2 +-
 ptx/sec_greensthm.ptx             |   8 +-
 ptx/sec_iterated_integrals.ptx    |   2 +-
 ptx/sec_line_int_intro.ptx        |   2 +-
 ptx/sec_line_int_vf.ptx           |  44 ++++-----
 ptx/sec_parametric_surfaces.ptx   |  26 ++---
 ptx/sec_stokes_divergence.ptx     |   2 +-
 ptx/sec_surface_integral.ptx      |   2 +-
 ptx/sec_triple_int.ptx            |   2 +-
 ptx/sec_vector_fields.ptx         |   4 +-
 13 files changed, 133 insertions(+), 130 deletions(-)

diff --git a/ptx/sec_center_of_mass.ptx b/ptx/sec_center_of_mass.ptx
index 54e787a83..f89df3a6b 100644
--- a/ptx/sec_center_of_mass.ptx
+++ b/ptx/sec_center_of_mass.ptx
@@ -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="4">
       <p>
         <em>Mass</em> and <em>weight</em> are different measures.
         Since they are scalar multiples of each other,
@@ -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="-1">
           <caption>A region <m>R</m> representing a lamina in <xref ref="ex_mass1"/></caption>
           <!-- START figures/fig_mass1.tex -->
           <image width="47%">
@@ -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%">
diff --git a/ptx/sec_cylindrical_spherical.ptx b/ptx/sec_cylindrical_spherical.ptx
index 5a90d7bb0..cbf199328 100644
--- a/ptx/sec_cylindrical_spherical.ptx
+++ b/ptx/sec_cylindrical_spherical.ptx
@@ -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"/>
@@ -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>
@@ -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>
@@ -728,7 +729,7 @@
       <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,
@@ -736,62 +737,7 @@
       </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>
@@ -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"/>
@@ -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>
@@ -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>
@@ -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;
diff --git a/ptx/sec_double_int_polar.ptx b/ptx/sec_double_int_polar.ptx
index 6d47f3cd9..0a58c6781 100644
--- a/ptx/sec_double_int_polar.ptx
+++ b/ptx/sec_double_int_polar.ptx
@@ -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 -->
@@ -619,7 +619,7 @@
 
             size(200,200,IgnoreAspect);
             //currentprojection=perspective(7,2,1);
-            currentprojection=orthographic(8,-13,2.8);
+            currentprojection=orthographic(8,4,2.8);
             defaultrender.merge=true;
 
             // setup and draw the axes
@@ -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.
@@ -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>
diff --git a/ptx/sec_double_int_volume.ptx b/ptx/sec_double_int_volume.ptx
index a2de4cef3..4393c482f 100644
--- a/ptx/sec_double_int_volume.ptx
+++ b/ptx/sec_double_int_volume.ptx
@@ -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>
diff --git a/ptx/sec_greensthm.ptx b/ptx/sec_greensthm.ptx
index 8bc7aa146..cced29847 100644
--- a/ptx/sec_greensthm.ptx
+++ b/ptx/sec_greensthm.ptx
@@ -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>
@@ -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>
@@ -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>
@@ -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>
diff --git a/ptx/sec_iterated_integrals.ptx b/ptx/sec_iterated_integrals.ptx
index c31635dde..020743a43 100644
--- a/ptx/sec_iterated_integrals.ptx
+++ b/ptx/sec_iterated_integrals.ptx
@@ -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%">
diff --git a/ptx/sec_line_int_intro.ptx b/ptx/sec_line_int_intro.ptx
index d0bc30c0c..4b64a6498 100644
--- a/ptx/sec_line_int_intro.ptx
+++ b/ptx/sec_line_int_intro.ptx
@@ -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>
diff --git a/ptx/sec_line_int_vf.ptx b/ptx/sec_line_int_vf.ptx
index 0000709ca..dda0287f0 100644
--- a/ptx/sec_line_int_vf.ptx
+++ b/ptx/sec_line_int_vf.ptx
@@ -741,7 +741,7 @@
           Evaluate <m>\int_C\vec F\cdot d\vec r</m>.
         </p>
 
-        <figure xml:id="fig_livf4" vshift="5">
+        <figure xml:id="fig_livf4" vshift="-2">
           <caption>The graph of <m>\vec r(t)</m> in <xref ref="ex_livf4"/></caption>
           <image width="47%">
             <shortdescription>A three-dimensional vector field and a portion of a helix curve.</shortdescription>
@@ -934,27 +934,6 @@
       though again it consists of two simply connected subregions.
     </p>
 
-    <p>
-      We have applied these terms to regions of the plane,
-      but they can be extended intuitively to domains in space
-      (and hyperspace).
-      In <xref ref="fig_simply_connected_spacea_3D"/>,
-      the domain bounded by the sphere
-      (at left)
-      and the domain with a subsphere removed
-      (at right)
-      are both simply connected.
-      Any simple closed path that lies entirely within these domains can be continuously deformed into a single point.
-      In <xref ref="fig_simply_connected_spacea_3D"/>,
-      neither domain is simply connected.
-      A left, the ball has a hole that extends its length and the pictured closed path cannot be deformed to a point.
-      At right, two paths are illustrated on the torus that cannot be shrunk to a point.
-    </p>
-
-    <p>
-      We will use the terms connected and simply connected in subsequent definitions and theorems.
-    </p>
-
     <figure xml:id="fig_simply_connected_plane" vshift="0">
       <caption><m>R_1</m> is simply connected; <m>R_2</m> is connected, but not simply connected; <m>R_3</m> is not connected</caption>
       <image width="47%">
@@ -1019,6 +998,27 @@
       </image>
     </figure>
 
+    <p>
+      We have applied these terms to regions of the plane,
+      but they can be extended intuitively to domains in space
+      (and hyperspace).
+      In <xref ref="fig_simply_connected_spacea_3D"/>,
+      the domain bounded by the sphere
+      (at left)
+      and the domain with a subsphere removed
+      (at right)
+      are both simply connected.
+      Any simple closed path that lies entirely within these domains can be continuously deformed into a single point.
+      In <xref ref="fig_simply_connected_spacea_3D"/>,
+      neither domain is simply connected.
+      A left, the ball has a hole that extends its length and the pictured closed path cannot be deformed to a point.
+      At right, two paths are illustrated on the torus that cannot be shrunk to a point.
+    </p>
+
+    <p>
+      We will use the terms connected and simply connected in subsequent definitions and theorems.
+    </p>
+
     <figure xml:id="fig_simply_connected_space">
       <caption>The domains in (a) are simply connected, while the domains in (b) are not</caption>
       <sidebyside widths="47% 47%" margins="0%">
diff --git a/ptx/sec_parametric_surfaces.ptx b/ptx/sec_parametric_surfaces.ptx
index ca249a81f..70aee076d 100644
--- a/ptx/sec_parametric_surfaces.ptx
+++ b/ptx/sec_parametric_surfaces.ptx
@@ -29,7 +29,7 @@
       Once this skill is developed,
       we'll show how to find the surface area <m>S</m> of a parametrically-defined surface <m>\surfaceS</m>, a skill needed in the remaining sections of this chapter.
     </p>
-    <aside vshift="0">
+    <aside vshift="4">
       <p>
         We use the letter <m>S</m> to denote Surface Area.
         This section begins a study into surfaces,
@@ -92,6 +92,17 @@
       </statement>
     </definition>
 
+    <aside vshift="2">
+      <p>
+        Recall that function is <em>one to one</em>
+        on its domain if the function never repeats an output value over the domain.
+        In the case of <m>\vec r(u,v)</m>,
+        <m>\vec r</m> is one to one if <m>\vec r(u_1,v_1) \neq \vec r(u_2,v_2)</m> for all points
+        <m>(u_1,v_1) \neq (u_2,v_2)</m> in the domain of <m>\vec r</m>.
+            <idx><h>one to one</h></idx>
+      </p>
+    </aside>
+
     <p>
       Given a point <m>(u_0,v_0)</m> in the domain of a vector-valued function <m>\vec r</m>,
       the vectors <m>\vec r_u(u_0,v_0)</m> and
@@ -104,17 +115,6 @@
       <m>\vec r_v</m> determine a plane that is tangent to <m>\surfaceS</m>.
     </p>
 
-    <aside vshift="0">
-      <p>
-        Recall that function is <em>one to one</em>
-        on its domain if the function never repeats an output value over the domain.
-        In the case of <m>\vec r(u,v)</m>,
-        <m>\vec r</m> is one to one if <m>\vec r(u_1,v_1) \neq \vec r(u_2,v_2)</m> for all points
-        <m>(u_1,v_1) \neq (u_2,v_2)</m> in the domain of <m>\vec r</m>.
-            <idx><h>one to one</h></idx>
-      </p>
-    </aside>
-
     <p>
       A surface <m>\surfaceS</m> is said to be <em>orientable</em>
           <idx><h>orientable</h></idx>
@@ -154,7 +154,7 @@
       The Möbius band is a non-orientable surface.
     </p>
 
-    <figure xml:id="fig_mobius" vshift="0">
+    <figure xml:id="fig_mobius" vshift="2">
       <caption>A Möbius band, a non-orientable surface</caption>
       <image width="47%">
         <shortdescription>A depction of a Möbius band, showing how the normal vector changes direction while traveling around the surface.</shortdescription>
diff --git a/ptx/sec_stokes_divergence.ptx b/ptx/sec_stokes_divergence.ptx
index 273dc7179..74f2b0081 100644
--- a/ptx/sec_stokes_divergence.ptx
+++ b/ptx/sec_stokes_divergence.ptx
@@ -42,7 +42,7 @@
       </statement>
     </theorem>
 
-    <aside vshift="-1">
+    <aside vshift="2">
       <p>
         <em>Note:</em> the term <q>outer unit normal vector</q>
         used in <xref ref="thm_divergence2"/>
diff --git a/ptx/sec_surface_integral.ptx b/ptx/sec_surface_integral.ptx
index 72cf716ad..c619156bc 100644
--- a/ptx/sec_surface_integral.ptx
+++ b/ptx/sec_surface_integral.ptx
@@ -342,7 +342,7 @@
           Find the flux of <m>\vec F</m> across <m>\surfaceS</m>.
         </p>
 
-        <figure xml:id="fig_surfflux1" vshift="0">
+        <figure xml:id="fig_surfflux1" vshift="2">
           <caption>The surface and vector field used in  <xref ref="ex_surfflux1"/></caption>
           <image width="47%">
             <shortdescription>A triangular surface in space, and a vector field flowing across the surface.</shortdescription>
diff --git a/ptx/sec_triple_int.ptx b/ptx/sec_triple_int.ptx
index ca832fc06..a783d4bda 100644
--- a/ptx/sec_triple_int.ptx
+++ b/ptx/sec_triple_int.ptx
@@ -613,7 +613,7 @@
           using the order of integration <m>dz\, dy\, dx</m>.
           Set up the triple integrals that give the volume in the other 5 orders of integration.
         </p>
-        <figure xml:id="fig_trip2a1_3D" vshift="-4">
+        <figure xml:id="fig_trip2a1_3D" vshift="-3">
           <caption>The region <m>D</m> used in <xref ref="ex_trip2"/></caption>
 
           <!-- START figures/figtrip2_3D.asy -->
diff --git a/ptx/sec_vector_fields.ptx b/ptx/sec_vector_fields.ptx
index 0ee61ef8f..fcc92631d 100644
--- a/ptx/sec_vector_fields.ptx
+++ b/ptx/sec_vector_fields.ptx
@@ -764,7 +764,7 @@
       The third component is the measure of curl of a planar vector field.
     </p>
 
-    <aside vshift="0">
+    <aside vshift="-2">
       <p>
         Note that in two dimensions, curl is a <em>scalar</em> quantity,
         while in three dimensions, curl is a vector quantity.
@@ -1365,7 +1365,7 @@
           <m>\divv \vec F = 0</m> and <m>\curl \vec F = \vec 0</m>.
         </p>
 
-        <figure xml:id="fig_vectorfield3" vshift="2">
+        <figure xml:id="fig_vectorfield3" vshift="4">
           <caption>A vector field representing a planar gravitational force</caption>
               <image width="47%">
                 <shortdescription>A radial vector field. Each vector points toward the origin, and vectors near the origin are larger.</shortdescription>

From 00b6f58effb0419f770fb3ce9eb008b28e9b6624 Mon Sep 17 00:00:00 2001
From: Gregory Hartman <apexcalculus@gmail.com>
Date: Fri, 14 Aug 2026 19:09:21 +0000
Subject: [PATCH 2/2] Final edits to chapters 13, 14 and 15

---
 ptx/sec_center_of_mass.ptx   | 6 +++---
 ptx/sec_double_int_polar.ptx | 2 +-
 2 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/ptx/sec_center_of_mass.ptx b/ptx/sec_center_of_mass.ptx
index f89df3a6b..8a59411a9 100644
--- a/ptx/sec_center_of_mass.ptx
+++ b/ptx/sec_center_of_mass.ptx
@@ -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="4">
+    <aside vshift="3">
       <p>
         <em>Mass</em> and <em>weight</em> are different measures.
         Since they are scalar multiples of each other,
@@ -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%">
@@ -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 -->
diff --git a/ptx/sec_double_int_polar.ptx b/ptx/sec_double_int_polar.ptx
index 0a58c6781..d7f8f00fa 100644
--- a/ptx/sec_double_int_polar.ptx
+++ b/ptx/sec_double_int_polar.ptx
@@ -619,7 +619,7 @@
 
             size(200,200,IgnoreAspect);
             //currentprojection=perspective(7,2,1);
-            currentprojection=orthographic(8,4,2.8);
+            currentprojection=orthographic(13,3,3);
             defaultrender.merge=true;
 
             // setup and draw the axes