updates

quarto/precalc/variables.qmd

@@ -111,6 +111,26 @@ a = v0 * cosd(theta)
 By defining a new variable `a` to represent a value that is repeated a few times in the expression, the last command is greatly simplified. Doing so makes it much easier to check for accuracy against the expression to compute.
 
 
+
+
+##### Example
+
+
+A common expression in mathematics is a polynomial expression, for example $-16s^2 + 32s - 12$. Translating this to `Julia` at $s =3$ we might have:
+
+
+```{julia}
+s = 3
+-16*s^2 + 32*s - 12
+```
+
+This looks nearly identical to the mathematical expression, but we inserted `*` to indicate multiplication between the constant and the variable. In fact, this step is not needed as Julia allows numeric literals to have an implied multiplication:
+
+
+```{julia}
+-16s^2 + 32s - 12
+```
+
 ##### Example
 
 A [grass swale](https://stormwater.pca.state.mn.us/index.php?title=Design_criteria_for_dry_swale_(grass_swale)) is a design to manage surface water flow resulting from a storm. Swales detain, filter, and infiltrate runoff limiting erosion in the process.
@@ -158,25 +178,6 @@ R = A / P
 Q = R^(2/3) * S^(1/2) / n * A
 ```
 
-
-##### Example
-
-
-A common expression in mathematics is a polynomial expression, for example $-16s^2 + 32s - 12$. Translating this to `Julia` at $s =3$ we might have:
-
-
-```{julia}
-s = 3
--16*s^2 + 32*s - 12
-```
-
-This looks nearly identical to the mathematical expression, but we inserted `*` to indicate multiplication between the constant and the variable. In fact, this step is not needed as Julia allows numeric literals to have an implied multiplication:
-
-
-```{julia}
--16s^2 + 32s - 12
-```
-
 ## Where math and computer notations diverge
 
 
@@ -243,7 +244,7 @@ The `varinfo` function will list the variables currently defined in the main wor
 
 :::{.callout-warning}
 ## Warning
-**Shooting oneselves in the foot.** `Julia` allows us to locally redefine variables that are built in, such as the value for `pi` or the function object assigned to `sin`. This is called shadowing. For example, this is a perfectly valid command `sin=3`. However, it will overwrite the typical value of `sin` so that `sin(3)` will be an error. At the terminal, the binding to `sin` occurs in the `Main` module. This shadows that value of `sin` bound in the `Base` module. Even if redefined in `Main`, the value in base can be used by fully qualifying the name, as in `Base.sin(pi)`. This uses the notation `module_name.variable_name` to look up a binding in a module.
+**Shooting oneselves in the foot.** `Julia` allows us to locally redefine variables that are built in, such as the value for `pi` or the function object assigned to `sin`. This is called shadowing. For example, this is a perfectly valid command `x + y = 3`. However, it doesn't specify an equation, rather it *redefines* addition. At the terminal, this binding to `+` occurs in the `Main` module. This shadows that value of `+` bound in the `Base` module. Even if redefined in `Main`, the value in base can be used by fully qualifying the name, as in `Base.:+(2, 3)`. This uses the notation `module_name.variable_name` to look up a binding in a module.
 
 :::
 
@@ -265,7 +266,7 @@ __private = 2     # a convention
 ### Unicode names
 
 
-Julia allows variable names to use Unicode identifiers. Such names allow `julia` notation to mirror that of many mathematical texts. For example, in calculus the variable $\epsilon$ is often used to represent some small number. We can assign to a symbol that looks like $\epsilon$ using `Julia`'s LaTeX input mode. Typing `\epsilon[tab]` will replace the text with the symbol within `IJulia` or the command line.
+`Julia` allows variable names to use Unicode identifiers. Such names allow `julia` notation to mirror that of many mathematical texts. For example, in calculus the variable $\epsilon$ is often used to represent some small number. We can assign to a symbol that looks like $\epsilon$ using `Julia`'s LaTeX input mode. Typing `\epsilon[tab]` will replace the text with the symbol within `IJulia` or the command line.
 
 
 ```{julia}