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jverzani
2025-07-27 15:26:00 -04:00
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10
quarto/differentiable_vector_calculus/vector_fields.qmd
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@@ -111,7 +111,7 @@ Plot of a vector field from $R^2 \rightarrow R^2$ illustrated by drawing curves
To the plot, we added the partial derivatives with respect to $r$ (in red) and with respect to $\theta$ (in blue). These are found with the soon-to-be discussed Jacobian. From the graph, you can see that these vectors are tangent vectors to the drawn curves.
The curves form a non-rectangular grid. Were the cells exactly parallelograms, the area would be computed taking into account the length of the vectors and the angle between them -- the same values that come out of a cross product.
The curves form a non-rectangular grid. Were the cells exactly parallelograms, the area would be computed taking into account the length of the vectors and the angle between them---the same values that come out of a cross product.
## Parametrically defined surfaces
@@ -323,7 +323,7 @@ plt = plot_axes()
We are using the vector of tuples interface (representing points) to specify the curve to draw.
Now we add on some curves for fixed $t$ and then fixed $\theta$ utilizing the fact that `project` returns a tuple of $x$--$y$ values to display.
Now we add on some curves for fixed $t$ and then fixed $\theta$ utilizing the fact that `project` returns a tuple of $x$---$y$ values to display.
```{julia}
for t in range(t₀, tₙ, 20)
@@ -1225,9 +1225,9 @@ q = interpolate(vcat(basic_conditions, new_conds))
plot_q_level_curve(q;layout=(1,2))
```
For this shape, if $b$ increases away from $b_0$, the secant line connecting $(a_0,0)$ and $(b, f(b)$ will have a negative slope, but there are no points nearby $x=c_0$ where the derivative has a tangent line with negative slope, so the continuous function is only on the left side of $b_0$. Mathematically, as $f$ is increasing $c_0$ -- as $f'''(c_0) = 3 > 0$ -- and $f$ is decreasing at $f(b_0)$ -- as $f'(b_0) = -1 < 0$, the signs alone suggest the scenario. The contour plot reveals, not one, but two one-sided functions of $b$ giving $c$.
For this shape, if $b$ increases away from $b_0$, the secant line connecting $(a_0,0)$ and $(b, f(b)$ will have a negative slope, but there are no points nearby $x=c_0$ where the derivative has a tangent line with negative slope, so the continuous function is only on the left side of $b_0$. Mathematically, as $f$ is increasing $c_0$---as $f'''(c_0) = 3 > 0$---and $f$ is decreasing at $f(b_0)$---as $f'(b_0) = -1 < 0$, the signs alone suggest the scenario. The contour plot reveals, not one, but two one-sided functions of $b$ giving $c$.
----
---
Now to characterize all possibilities.
@@ -1291,7 +1291,7 @@ $$
Then $F(c, b) = g_1(b) - g_2(c)$.
By construction, $g_2(c_0) = 0$ and $g_2^{(k)}(c_0) = f^{(k+1)}(c_0)$,
Adjusting $f$ to have a vanishing second -- but not third -- derivative at $c_0$ means $g_2$ will satisfy the assumptions of the lemma assuming $f$ has at least four continuous derivatives (as all our example polynomials do).
Adjusting $f$ to have a vanishing second---but not third---derivative at $c_0$ means $g_2$ will satisfy the assumptions of the lemma assuming $f$ has at least four continuous derivatives (as all our example polynomials do).
As for $g_1$, we have by construction $g_1(b_0) = 0$. By differentiation we get a pattern for some constants $c_j = (j+1)\cdot(j+2)\cdots \cdot k$ with $c_k = 1$.