make pdf file generation work

quarto/ODEs/euler.qmd

@@ -70,7 +70,6 @@ That is, if we stitched together pieces of the slope field, would we get a curve
 ```{julia}
 #| hold: true
 #| echo: false
-#| cache: true
 ## {{{euler_graph}}}
 function make_euler_graph(n)
     x, y = symbols("x, y")
@@ -241,17 +240,7 @@ It is more work for the computer, but not for us, and clearly a much better appr
 
 ## The Euler method
 
-
-```{julia}
-#| hold: true
-#| echo: false
-imgfile ="figures/euler.png"
-caption = """
-Figure from first publication of Euler's method. From [Gander and Wanner](http://www.unige.ch/~gander/Preprints/Ritz.pdf).
-"""
-
-ImageFile(:ODEs, imgfile, caption)
-```
+![Figure from first publication of Euler's method. From [Gander and Wanner](http://www.unige.ch/~gander/Preprints/Ritz.pdf).](./figures/euler.png)
 
 The name of our function reflects the [mathematician](https://en.wikipedia.org/wiki/Leonhard_Euler) associated with the iteration:
 
@@ -361,9 +350,13 @@ caption = """
 A child's bead game. What shape wire will produce the shortest time for a bed to slide from a top to the bottom?
 
 """
-ImageFile(:ODEs, imgfile, caption)
+#ImageFile(:ODEs, imgfile, caption)
+nothing
 ```
 
+![A child's bead game. What shape wire will produce the shortest time for a bed to slide from a top to the bottom?](./figures/bead-game.jpg)
+
+
 Restrict our attention to the $x$-$y$ plane, and consider a path, between the point $(0,A)$ and $(B,0)$. Let $y(x)$ be the distance from $A$, so $y(0)=0$ and at the end $y$ will be $A$.
 
 
@@ -378,16 +371,22 @@ caption = """
 As early as 1638, Galileo showed that an object falling along `AC` and then `CB` will fall faster than one traveling along `AB`, where `C` is on the arc of a circle.
 From the [History of Math Archive](http://www-history.mcs.st-and.ac.uk/HistTopics/Brachistochrone.html).
 """
-ImageFile(:ODEs, imgfile, caption)
+#ImageFile(:ODEs, imgfile, caption)
+nothing
 ```
 
+![As early as 1638, Galileo showed that an object falling along `AC`
+and then `CB` will fall faster than one traveling along `AB`, where
+`C` is on the arc of a circle.  From the [History of Math
+Archive](http://www-history.mcs.st-and.ac.uk/HistTopics/Brachistochrone.html).
+](./figures/galileo.png)
+
 This simulation also suggests that a curved path is better than the shorter straight one:
 
 
 ```{julia}
 #| hold: true
 #| echo: false
-#| cache: true
 ##{{{brach_graph}}}
 
 function brach(f, x0, vx0, y0, vy0, dt, n)
@@ -603,13 +602,11 @@ $$
 We can try the Euler method here. A simple approach might be this iteration scheme:
 
 
-$$
 \begin{align*}
 x_{n+1} &= x_n + h,\\
 u_{n+1} &= u_n + h v_n,\\
 v_{n+1} &= v_n - h \cdot g/l \cdot \sin(u_n).
 \end{align*}
-$$
 
 Here we need *two* initial conditions: one for the initial value $u(t_0)$ and the initial value of $u'(t_0)$. We have seen if we start at an angle $a$ and release the bob from rest, so $u'(0)=0$ we get a sinusoidal answer to the linearized model. What happens here? We let $a=1$, $L=5$ and $g=9.8$: