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2026 summer edits chpt12 #449

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3333ba9
Initial commit.
APEXCalculus Aug 12, 2026
60defcf
Adjustments to Chapter 12, mostly vertical places of figures.
APEXCalculus Aug 13, 2026
f9f744f
Fixed an aside
APEXCalculus Aug 13, 2026
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48 changes: 21 additions & 27 deletions ptx/sec_curvature.ptx
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Expand Up @@ -208,7 +208,7 @@
We find it with <m>\vec r(2) = \la 1/5, 18/5\ra</m>.
</p>

<figure xml:id="fig_vvfarc1" vshift="-2">
<figure xml:id="fig_vvfarc1" vshift="0">
<caption>Graphing <m>\vec r</m> in <xref ref="ex_vvfarc1"/> with parameters <m>t</m> and <m>s</m></caption>
<!-- START figures/fig_vvfarc1.tex -->
<image width="47%">
Expand Down Expand Up @@ -687,7 +687,7 @@
Being able to think of curvature in terms of the radius of a circle is very useful.
</p>

<figure xml:id="fig_curvature_osculate" vshift="6">
<figure xml:id="fig_curvature_osculate" vshift="3">
<caption>Illustrating the osculating circles for the curve seen in <xref ref="fig_curvature_intro"/></caption>
<!-- START figures/fig_curvature_introc.tex -->
<image width="47%">
Expand Down Expand Up @@ -757,7 +757,7 @@
</md>.
</p>

<figure xml:id="fig_curvature3" vshift="0">
<figure xml:id="fig_curvature3" vshift="1">
<caption>Examining the curvature of <m>y=x^2</m></caption>
<!-- START figures/fig_curvature3.tex -->
<image width="47%">
Expand Down Expand Up @@ -834,26 +834,10 @@
</md>
</p>

<p>
While this is not a particularly <q>nice</q> formula,
it does explicitly tell us what the curvature is at a given <m>t</m> value.
To maximize <m>\kappa(t)</m>,
we should solve <m>\kappa'(t)=0</m> for <m>t</m>.
This is doable, but <em>very</em> time consuming.
Instead, consider the graph of
<m>\kappa(t)</m> as given in <xref ref="fig_curvature4a"/>.
We see that <m>\kappa</m> is maximized at two <m>t</m> values;
using a numerical solver, we find these values are <m>t\approx\pm 0.189</m>.
In <xref ref="fig_curvature4b_3D"/> we graph <m>\vrt</m> and indicate the points where curvature is maximized.
</p>

<figure xml:id="fig_curvature4">
<caption>Understanding the curvature of a curve in space</caption>
<sidebyside widths="47% 47%" margins="0%" valign="bottom">
<figure xml:id="fig_curvature4a">
<caption>The curvature of <m>\vec{r}(t)</m></caption>
<figure xml:id="fig_curvature4a" vshift="4">
<caption>The curvature of <m>\vec{r}(t)</m> in <xref ref="ex_curvature4"/></caption>
<!-- START figures/fig_curvature4a.tex -->
<image>
<image width="47%">
<shortdescription>A plot of the curvature as a function of the parameter t.</shortdescription>
<description>
<p>
Expand Down Expand Up @@ -885,11 +869,11 @@
<!-- figures/fig_curvature4a.tex END -->
</figure>

<figure xml:id="fig_curvature4b_3D">
<caption>A plot of the curve <m>\vec{r}(t)=\la t, t^2, 2t^3\ra</m></caption>
<figure xml:id="fig_curvature4b_3D" vshift="-3">
<caption>A plot of the curve <m>\vec{r}(t)=\la t, t^2, 2t^3\ra</m> in <xref ref="ex_curvature4"/></caption>

<!-- START figures/figcurvature4_3D.asy -->
<image>
<image width="47%">
<shortdescription>A plot of the vector-valued function in this example, with points of maximum curvature marked.</shortdescription>
<description>
<p>
Expand Down Expand Up @@ -950,9 +934,19 @@
</image>
<!-- figures/figcurvature4_3D.asy END -->
</figure>
</sidebyside>

</figure>
<p>
While this is not a particularly <q>nice</q> formula,
it does explicitly tell us what the curvature is at a given <m>t</m> value.
To maximize <m>\kappa(t)</m>,
we should solve <m>\kappa'(t)=0</m> for <m>t</m>.
This is doable, but <em>very</em> time consuming.
Instead, consider the graph of
<m>\kappa(t)</m> as given in <xref ref="fig_curvature4a"/>.
We see that <m>\kappa</m> is maximized at two <m>t</m> values;
using a numerical solver, we find these values are <m>t\approx\pm 0.189</m>.
In <xref ref="fig_curvature4b_3D"/> we graph <m>\vrt</m> and indicate the points where curvature is maximized.
</p>

</solution>
</example>
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44 changes: 20 additions & 24 deletions ptx/sec_tan_norm.ptx
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Expand Up @@ -68,7 +68,7 @@
since they are only length 1.)
</p>

<figure xml:id="fig_tannorm1" vshift="-1">
<figure xml:id="fig_tannorm1" vshift="2">
<caption>Plotting unit tangent vectors in <xref ref="ex_tannorm1"/></caption>

<!-- START figures/figtannorm1_3D.asy -->
Expand Down Expand Up @@ -382,7 +382,7 @@
These are sketched in <xref ref="fig_tannorm3"/>.
</p>

<figure xml:id="fig_tannorm3" vshift="0">
<figure xml:id="fig_tannorm3" vshift="3">
<caption>Plotting unit tangent and normal vectors in <xref ref="ex_tannorm3"/></caption>

<!-- START figures/figtannorm3_3D.asy -->
Expand Down Expand Up @@ -447,17 +447,7 @@
</solution>
</example>

<aside vshift="-6">
<p>
There is one flaw in our definition of <m>\unitnormal(t)</m>:
it is possible that we could have <m>\unittangentprime(t)=0</m>!
Indeed, this is the case for any line of the form
<m>\vec{\ell}(t) = \vec{r}_0+t\vec{v}</m>.
For straight lines in the plane,
it is most common to orient the normal vector <m>90^\circ</m> counterclockwise from the tangent vector,
but for lines in three dimensions, there is no preferred choice of normal vector.
</p>
</aside>


<p>
The previous example was once again
Expand Down Expand Up @@ -517,7 +507,7 @@
we compute the unit tangent and normal vectors for <m>t=-1,0</m> and 1 and sketch them in <xref ref="fig_tannorm4"/>.
</p>

<figure xml:id="fig_tannorm4" vshift="-5">
<figure xml:id="fig_tannorm4" vshift="0">
<caption>Plotting unit tangent and normal vectors in <xref ref="ex_tannorm4"/></caption>
<!-- START figures/fig_tannorm4.tex -->
<image width="47%">
Expand Down Expand Up @@ -607,6 +597,18 @@
</p>
</statement>
</theorem>

<p>
A brief consideration of this theorem may make one wonder: what if the graph of <m>\vec r</m> does
not have a concave side? What if <m>\vec r</m> is a line?
</p>
<p>
This exposes a shortcoming in our definition of <m>\unitnormal(t)</m>, where
we require that <m>\unittangent(t)</m> be smooth, i.e., that <m>\unittangentprime(t) \neq \vec 0</m>,
a requirement that lines do not fulfill. One may still want to compute a normal vector for a given line, though.
For straight lines in the <m>x,y</m> plane, it is most common to orient the normal vector <m>90^\circ</m>
counterclockwise from the tangent vector. For lines in three dimensions, there is no preferred choice of normal vector.
</p>
</subsection>

<subsection>
Expand Down Expand Up @@ -802,7 +804,7 @@
gives a graph of the path for reference.
</p>

<figure xml:id="fig_tannorm6" vshift="0">
<figure xml:id="fig_tannorm6" vshift="2">
<caption>Graphing <m>\vec r(t)</m> in <xref ref="ex_tannorm6"/></caption>
<!-- START figures/fig_tannorm6.tex -->
<image width="47%">
Expand Down Expand Up @@ -875,7 +877,7 @@
which we plot in <xref ref="fig_tannorm7b"/>.
</p>

<figure xml:id="fig_tannorm7b" vshift="-2">
<figure xml:id="fig_tannorm7b" vshift="0">
<caption>Plotting the position of a thrown ball, with 1s increments shown</caption>
<!-- START figures/fig_tannorm7b.tex -->
<image width="47%">
Expand Down Expand Up @@ -1043,10 +1045,7 @@
<statement>
<p>
If <m>\unittangent(t)</m> is a unit tangent vector,
what is <m>\norm{\unittangent(t)}</m>?
</p>

<p>
what is <m>\norm{\unittangent(t)}</m>?
<fillin answer="1" width="5"/>
</p>
</statement>
Expand All @@ -1072,10 +1071,7 @@
<statement>
<p>
If <m>\unitnormal(t)</m> is a unit normal vector,
what is <m>\unitnormal(t)\cdot \vrp(t)</m>?
</p>

<p>
what is <m>\unitnormal(t)\cdot \vrp(t)</m>?
<fillin answer="0" width="5"/>
</p>
</statement>
Expand Down
4 changes: 2 additions & 2 deletions ptx/sec_vvf.ptx
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Expand Up @@ -648,7 +648,7 @@
<m>\vec r(t) = \vec p(t) + \vec c(t) = \la \cos(t) + t,-\sin(t) +1\ra</m>,
which is graphed in <xref ref="fig_vvf4b"/>.
</p>
<figure xml:id="fig_vvf4b" vshift="2">
<figure xml:id="fig_vvf4b" vshift="3">
<caption>The cycloid in <xref ref="ex_vvf4"/></caption>
<!-- START figures/fig_vvf4a.tex -->
<image width="47%">
Expand Down Expand Up @@ -750,7 +750,7 @@
The displacement of <m>\vec r(t)</m> on <m>[-1,1]</m> is thus <m>\vec d = \la 0,1\ra - \la 0,-1\ra = \la 0,2\ra</m>.
</p>

<figure xml:id="fig_vvf5" vshift="-6">
<figure xml:id="fig_vvf5" vshift="2">
<caption>Graphing the displacement of a position function in <xref ref="ex_vvf5"/></caption>
<!-- START figures/fig_vvf5.tex -->
<image width="47%">
Expand Down
29 changes: 15 additions & 14 deletions ptx/sec_vvf_calc.ptx
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Expand Up @@ -19,7 +19,7 @@
The theorem following the definition shows that in practice,
taking limits of vector-valued functions is no more difficult than taking limits of real-valued functions.
</p>
<aside vshift="0">
<aside vshift="-2">
<p>
We can define one-sided limits in a manner very similar to <xref ref="def_vvf_limit"/>.
</p>
Expand Down Expand Up @@ -339,13 +339,21 @@
</statement>
</definition>

<aside vshift="0">
<aside vshift="3">
<p>
Alternate notations for the derivative of <m>\vec r</m> include:
Two notes on <xref ref="def_vvf_derivative"/>:
</p>
<p>
First, alternate notations for the derivative of <m>\vec r</m> include:
<md>
\vrp(t) = \frac{d}{dt}\big(\,\vec r(t)\,\big) = \frac{d\vec r}{dt}
</md>.
</p>
<p>
Second, using one-sided limits,
we can define differentiability on closed intervals.
We'll make use of this a few times in this chapter.
</p>
</aside>
<p>
If a vector-valued function has a derivative for all <m>c</m> in an open interval <m>I</m>,
Expand All @@ -359,14 +367,7 @@
The following theorem verifies that this means we can compute derivatives component-wise as well,
making the task not too difficult.
</p>
<aside vshift="0">
<p>
Again, using one-sided limits,
we can define differentiability on closed intervals.
We'll make use of this a few times in this chapter.
</p>
</aside>


<theorem xml:id="thm_vvf_deriv">
<title>Derivatives of Vector-Valued Functions</title>
<statement>
Expand Down Expand Up @@ -992,7 +993,7 @@
<m>\vec u(t)</m> is 1 for all <m>t</m>.
</p>

<figure xml:id="fig_vvfderiv2a" vshift="-3">
<figure xml:id="fig_vvfderiv2a" vshift="0">
<caption>Graphing <m>\vec r(t)</m> and <m>\vec u(t)</m> in <xref ref="ex_vvfderiv2"/></caption>
<!-- START figures/fig_vvfderiv2a.tex -->
<image width="47%">
Expand Down Expand Up @@ -1070,7 +1071,7 @@
</md>
</p>

<figure xml:id="fig_vvfderiv2b" vshift="-5">
<figure xml:id="fig_vvfderiv2b" vshift="2">
<caption>Graphing some of the derivatives of <m>\vec u(t)</m> in <xref ref="ex_vvfderiv2"/></caption>
<!-- START figures/fig_vvfderiv2b.tex -->
<image width="47%">
Expand Down Expand Up @@ -2312,7 +2313,7 @@

<statement>
<p>
<m>\displaystyle\int_0^{\pi} \la -\sin(t) ,\cos(t) \ra\,dt=</m><var name="$int" width="30"/>.
<m>\displaystyle\int_0^{\pi} \la -\sin(t) ,\cos(t) \ra\,dt=</m><var name="$int" width="20"/>.
</p>
</statement>
</webwork>
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2 changes: 2 additions & 0 deletions ptx/sec_vvf_motion.ptx
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Expand Up @@ -863,6 +863,8 @@
</md>.
</p>

<p xml:id="pagebreak-projectile-motion"></p>

<insight xml:id="idea_projectile">
<title>Projectile Motion</title>
<p>
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1 change: 1 addition & 0 deletions publication/publication-pdf.ptx
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Expand Up @@ -47,6 +47,7 @@
<insertions pagebreaks="exset-double-polar-special exset-cylindrical-spherical-setup exset-Greens-divergence exset-vectors-balance-forces"/>
<!-- other page breaks -->
<insertions pagebreaks="pagebreak-taylor-series pagebreak-iterated-int pagebreak-glossary"/>
<insertions pagebreaks="pagebreak-projectile-motion"/>
</latex>
<epub>
<cover front="cover/cover.png"/>
Expand Down