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jverzani
2025-07-27 15:26:00 -04:00
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quarto/differentiable_vector_calculus/scalar_functions.qmd
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@@ -388,7 +388,7 @@ For a scalar function, Define a *level curve* as the solutions to the equations
contour(xsₛ, ysₛ, zzsₛ)
```
Were one to walk along one of the contour lines, then there would be no change in elevation. The areas of greatest change in elevation - basically the hills - occur where the different contour lines are closest. In this particular area, there is a river that runs from the upper right through to the lower left and this is flanked by hills.
Were one to walk along one of the contour lines, then there would be no change in elevation. The areas of greatest change in elevation---basically the hills--- occur where the different contour lines are closest. In this particular area, there is a river that runs from the upper right through to the lower left and this is flanked by hills.
The $c$ values for the levels drawn may be specified through the `levels` argument:
@@ -636,7 +636,7 @@ This says, informally, for any scale about $L$ there is a "ball" about $C$ (not
In the univariate case, it can be useful to characterize a limit at $x=c$ existing if *both* the left and right limits exist and the two are equal. Generalizing to getting close in $R^m$ leads to the intuitive idea of a limit existing in terms of any continuous "path" that approaches $C$ in the $x$-$y$ plane has a limit and all are equal. Let $\gamma$ describe the path, and $\lim_{s \rightarrow t}\gamma(s) = C$. Then $f \circ \gamma$ will be a univariate function. If there is a limit, $L$, then this composition will also have the same limit as $s \rightarrow t$. Conversely, if for *every* path this composition has the *same* limit, then $f$ will have a limit.
The "two path corollary" is a trick to show a limit does not exist - just find two paths where there is a limit, but they differ, then a limit does not exist in general.
The "two path corollary" is a trick to show a limit does not exist---just find two paths where there is a limit, but they differ, then a limit does not exist in general.
### Continuity of scalar functions
@@ -997,7 +997,7 @@ The figure suggests a potential geometric relationship between the gradient and
We see here how the gradient of $f$, $\nabla{f} = \langle f_{x_1}, f_{x_2}, \dots, f_{x_n} \rangle$, plays a similar role as the derivative does for univariate functions.
First, we consider the role of the derivative for univariate functions. The main characterization - the derivative is the slope of the line that best approximates the function at a point - is quantified by Taylor's theorem. For a function $f$ with a continuous second derivative:
First, we consider the role of the derivative for univariate functions. The main characterization---the derivative is the slope of the line that best approximates the function at a point---is quantified by Taylor's theorem. For a function $f$ with a continuous second derivative:
$$
@@ -1174,7 +1174,7 @@ atand(mean(slopes))
Which seems about right for a generally uphill trail section, as this is.
In the above example, the data is given in terms of a sample, not a functional representation. Suppose instead, the surface was generated by `f` and the path - in the $x$-$y$ plane - by $\gamma$. Then we could estimate the maximum and average steepness by a process like this:
In the above example, the data is given in terms of a sample, not a functional representation. Suppose instead, the surface was generated by `f` and the path---in the $x$-$y$ plane---by $\gamma$. Then we could estimate the maximum and average steepness by a process like this:
```{julia}