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claude typos; plotly+scatter3d issue; google analytics

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jverzani committed 2026-06-03 20:46:39 -04:00
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quarto/.gitignore
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@@ -5,4 +5,5 @@
/*/*.ipynb/
/*/bonepile.qmd
/*/references.bib
weave_support.jl
weave_support.jl
**/*.quarto_ipynb
quarto/_quarto.yml
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@@ -1,4 +1,4 @@
version: "0.25"
version: "0.27"
engines: ['julia']
project:
@@ -10,6 +10,7 @@ comments:
book:
title: "Calculus with Julia"
author: "John Verzani"
google-analytics: "G-LXFE6MTM4M"
date: now
search: true
repo-url: https://github.com/jverzani/CalculusWithJuliaNotes.jl
@@ -117,7 +118,7 @@ book:
- part: alternatives.qmd
chapters:
- alternatives/giac.qmd
#- alternatives/giac.qmd
- alternatives/symbolics.qmd
- alternatives/SciML.qmd
#- alternatives/interval_arithmetic.qmd
quarto/alternatives.qmd
-2
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@@ -2,8 +2,6 @@
These notes use a particular selection of packages. This selection could have been different. For example:
* The symbolic math is provided by `SymPy`. [Giac](./alternatives/giac.html) and
[Symbolics](./alternatives/symbolics.html) (along with `SymbolicUtils` and `ModelingToolkit`) provide alternatives.
* The finding of zeros of scalar-valued, univariate functions is done with `Roots`. The [NonlinearSolve](./alternatives/SciML.html#nonlinearsolve) package provides an alternative for univariate and multi-variate functions.
quarto/alternatives/symbolics.qmd
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@@ -1,6 +1,8 @@
# Symbolics.jl
XXX This needs updating! XXX
XXX add https://docs.sciml.ai/SymbolicIntegration/stable/ XXX
There are a few options in `Julia` for symbolic math, for example, the `SymPy` package which wraps a Python library. This section describes a collection of native `Julia` packages providing many features of symbolic math.
@@ -238,27 +240,6 @@ w = x^3 + y^3 - 2z^3
substitute(w, Dict(x=>2, y=>3))
```
The `fold` argument can be passed `false` to inhibit evaluation of values. Compare:
```{julia}
ex = 1 + sqrt(x)
substitute(ex, x=>2), substitute(ex, x=>2, fold=false)
```
Or
```{julia}
ex = sin(x)
substitute(ex, x=>π), substitute(ex, x=>π, fold=false)
```
For the latter, it is more efficient to directly use `Term`, which creates the symbolic expression representing the calling of `sin(π)`:
```{julia}
Symbolics.Term(sin, [π])
```
### Simplify
quarto/differentiable_vector_calculus/scalar_functions_applications-backup-question.qmd
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quarto/differentiable_vector_calculus/scalar_functions_applications.qmd
+34 -14
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@@ -9,7 +9,7 @@ This section uses these add-on packages:
```{julia}
using CalculusWithJulia
using Plots
plotly()
#plotly()
using SymPy
using Roots
```
@@ -33,7 +33,6 @@ Consider the case $f:R^2 \rightarrow R$. We visualize $z=f(x,y)$ through a surfa
For the univariate case, the tangent line has many different uses. Here we see the tangent plane also does.
### Equation of the tangent plane
@@ -119,6 +118,7 @@ arrow!(pt, [1,0,0], linestyle=:dash)
arrow!(pt, [0,1,0], linestyle=:dash)
```
#### Alternate forms
@@ -170,7 +170,6 @@ For clarity:
* The scalar function $z = f(x,y)$ describes a surface, $(x,y,f(x,y))$; the gradient, $\nabla{f}$, is $2$ dimensional and points in the direction of greatest ascent for the surface.
* The scalar function $f(x,y,z)$ *also* describes a surface, through level curves $f(x,y,z) = c$, for some *constant* $c$. The gradient $\nabla{f}$ is $3$ dimensional and *orthogonal* to the surface.
##### Example
@@ -231,6 +230,7 @@ t = 1/2
(n1(gamma(t)) × n2(gamma(t))) × gamma'(t)
```
#### Plotting level curves of $F(x,y,z) = c$
@@ -250,7 +250,6 @@ f(x,y,z) = (x^2 + ((1+b) * y)^2 + z^2 - 1)^3 - x^2 * z^3 - a * y^2 * z^3
CalculusWithJulia.plot_implicit_surface(f, xlim=-2..2, ylim=-1..1, zlim=-1..2)
```
## Linearization
@@ -329,7 +328,7 @@ Here, $\nabla{f}$ describes a *normal* to the tangent plane. The description of
f(x,y,z) = x^4 -x^3 + y^2 + z^2
f(v) = f(v...)
a, b,c = ∇(f)(2,2,2)
"$a x + $b y + $c z = $([a,b,c] ⋅ [2,2,2])"
println("$a x + $b y + $c z = $([a,b,c] ⋅ [2,2,2])");
```
### Newton's method to solve $f(x,y) = 0$ and $g(x,y)=0$.
@@ -504,6 +503,9 @@ function nm(f, g, x, n=5)
end
```
##### Example
@@ -655,6 +657,7 @@ plot(F, 0.01, 5) # p > 0
This problem does not have a readily expressed value for $x^*$, but when $p \approx 0$ we should get similar behavior to the intersection of $y=px$ and $y=\pi/2 - x$ for $x^*$, or $x^* \approx \pi/(2(1+p))$ which has derivative of $-\pi/2$ at $p=0$, matching the above graph. For *large* $p$, the problem looks like the intersection of the line $y=1$ with $y=px$ or $x^* \approx 1/p$ which has derivative that goes to $0$ as $p$ goes to infinity, again matching this graph.
## Optimization
@@ -807,7 +810,6 @@ det(H_a)
(The test is inconclusive, as it needs the function to "fall away" from the tangent plane in all directions, in this case, along a circular curve, the function touches the tangent plane, so it doesn't fall away.)
##### Example
@@ -853,6 +855,7 @@ end
p
```
##### Example
@@ -875,9 +878,8 @@ At $\vec{a}$ this has positive determinant and $f_{xx} > 0$, so $\vec{a}$ corres
```{julia}
fₗ(x,y) = x^2 + 2y^2 - x
fₗ(v) = fₗ(v...)
gammaₗ(t) = [cos(t), sin(t)] # traces out x^2 + y^2 = 1 over [0, 2pi]
gₗ = fₗ ∘ gammaₗ
gₗ = splat(fₗ) ∘ gammaₗ
cpsₗ = find_zeros(gₗ', 0, 2pi) # critical points of g
append!(cpsₗ, [0, 2pi])
@@ -907,6 +909,12 @@ So we have the maximum occurs at the angles $2\pi/3$ and $4\pi/3$. Here we visua
hₗ(x,y) = fₗ(x,y) * (x^2 + y^2 <= 1 ? 1 : NaN)
```
```{julia}
#| echo: false
gr(); # scatter3d! and plotly() are failing!!
```
```{julia}
#| hold: true
xs = ys = range(-1,1, length=100)
@@ -918,6 +926,11 @@ zs = fₗ.(xs, ys)
scatter3d!(xs, ys, zs)
```
```{julia}
#| echo: false
#plotly();
```
A contour plot also shows that some---and only one---extrema happens on the interior:
@@ -930,6 +943,7 @@ contour(xs, ys, hₗ)
The extrema are identified by the enclosing regions, in this case the one around the point $(1/2, 0)$.
##### Example: Steiner's problem
@@ -1189,6 +1203,10 @@ end
We now visualize, using the `Contour` package to draw the contour lines in the $x-y$ plane:
```{julia}
#| echo: false
gr()
```
```{julia}
#| hold: true
@@ -1217,6 +1235,11 @@ scatter3d!(unzip(pts)...)
plot!(unzip(pts)..., linewidth=3)
```
```{julia}
#| echo: false
#plotly()
```
### Newton's method for minimization
@@ -2324,7 +2347,6 @@ Which vector is orthogonal to the contour line $x^2 + y^2 = 3$?
```{julia}
#| hold: true
#| echo: false
choices = [
raw"`` \langle 2x, 2y\rangle``",
@@ -2339,7 +2361,6 @@ Due to the form of the gradient of the constraint, finding when $\nabla{f} = \la
```{julia}
#| hold: true
f(x,y) = exp(-x^2-y^2) * (2x^2 + y^2)
f(v) = f(v...)
r(t) = sqrt(3)*[cos(t), sin(t)]
@@ -2352,15 +2373,14 @@ Using these points, what is the largest value on the boundary?
```{julia}
#| hold: true
#| eval: false
#| echo: false
f(x,y) = exp(-x^2-y^2) * (2x^2 + y^2)
f(v) = f(v...)
r(t) = sqrt(3)*[cos(t), sin(t)]
rat(x) = abs(x[1]/x[2]) - 1
fn = rat ∘ ∇(f) ∘ r
fn = rat ∘ ∇(splat(f)) ∘ r
ts = fzeros(fn, 0, 2pi)
val = maximum((f∘r).(ts))
val = maximum((splat(u)∘r).(ts))
numericq(val)
```
quarto/differentiable_vector_calculus/test.html
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<div class="quarto-title">
<h1 class="title">Applications with scalar functions</h1>
</div>
<div class="quarto-title-meta">
</div>
</header>
<p>This section uses these add-on packages:</p>
<div id="14" class="cell" data-execution_count="1">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb1"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">CalculusWithJulia</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Plots</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plotly</span>()</span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">SymPy</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Roots</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
</div>
<section id="example" class="level5">
<h5 class="anchored" data-anchor-id="example">Example</h5>
<p>Consider the function <span class="math inline">\(f(x,y) = x^2 + 3y^2 -x\)</span> over the region <span class="math inline">\(x^2 + y^2 \leq 1\)</span>. This is a continuous function over a closed set, so will have both an absolute maximum and minimum. Find these from an investigation of the critical points and the boundary points.</p>
<p>The gradient is easily found: <span class="math inline">\(\nabla{f} = \langle 2x - 1, 6y \rangle\)</span>, and is <span class="math inline">\(\vec{0}\)</span> only at <span class="math inline">\(\vec{a} = \langle 1/2, 0 \rangle\)</span>. The Hessian is:</p>
<p><span class="math display">\[
H =
\begin{bmatrix}
2 &amp; 0\\
0 &amp; 6
\end{bmatrix}.
\]</span></p>
<p>At <span class="math inline">\(\vec{a}\)</span> this has positive determinant and <span class="math inline">\(f_{xx} &gt; 0\)</span>, so <span class="math inline">\(\vec{a}\)</span> corresponds to a <em>local</em> minimum with values <span class="math inline">\(f(\vec{a}) = (1/2)^2 + 3(0) - 1/2 = -1/4\)</span>. The absolute maximum and minimum may occur here (well, not the maximum) or on the boundary, so that must be considered. In this case we can easily parameterize the boundary and turn this into the univariate case:</p>
<div id="16" class="cell" data-execution_count="1">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb2"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">fₗ</span>(x,y) <span class="op">=</span> x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> <span class="fl">2</span>y<span class="op">^</span><span class="fl">2</span> <span class="op">-</span> x</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="fu">gammaₗ</span>(t) <span class="op">=</span> [<span class="fu">cos</span>(t), <span class="fu">sin</span>(t)] <span class="co"># traces out x^2 + y^2 = 1 over [0, 2pi]</span></span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>gₗ <span class="op">=</span> <span class="fu">splat</span>(fₗ) <span class="op">∘</span> gammaₗ</span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a>cpsₗ <span class="op">=</span> <span class="fu">find_zeros</span>(gₗ<span class="ch">', 0, 2pi) # critical points of g</span></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a><span class="fu">append!</span>(cpsₗ, [<span class="fl">0</span>, <span class="fl">2</span>pi])</span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a><span class="fu">unique!</span>(cpsₗ)</span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a><span class="fu">gₗ</span>.(cpsₗ)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-display" data-execution_count="1">
<pre><code>5-element Vector{Float64}:
0.0
2.25
2.0
2.25
0.0</code></pre>
</div>
</div>
<p>We see that maximum value is <code>2.25</code> and that the interior point, <span class="math inline">\(\vec{a}\)</span>, will be where the minimum value occurs. To see exactly where the maximum occurs, we look at the values of gamma:</p>
<div id="18" class="cell" data-execution_count="1">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb4"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>inds <span class="op">=</span> [<span class="fl">2</span>,<span class="fl">4</span>]</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>cpsₗ[inds]</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-display" data-execution_count="1">
<pre><code>2-element Vector{Float64}:
2.0943951023931953
4.1887902047863905</code></pre>
</div>
</div>
<p>These are multiples of <span class="math inline">\(\pi\)</span>:</p>
<div id="20" class="cell" data-execution_count="1">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb6"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a>cpsₗ[inds]<span class="op">/</span><span class="cn">pi</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-display" data-execution_count="1">
<pre><code>2-element Vector{Float64}:
0.6666666666666666
1.3333333333333333</code></pre>
</div>
</div>
<p>So we have the maximum occurs at the angles <span class="math inline">\(2\pi/3\)</span> and <span class="math inline">\(4\pi/3\)</span>. Here we visualize, using a hacky trick of assigning <code>NaN</code> values to the function to avoid plotting outside the circle:</p>
<div id="22" class="cell" data-execution_count="1">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb8"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="fu">hₗ</span>(x,y) <span class="op">=</span> <span class="fu">fₗ</span>(x,y) <span class="op">*</span> (x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> y<span class="op">^</span><span class="fl">2</span> <span class="op">&lt;=</span> <span class="fl">1</span> ? <span class="fl">1</span> <span class="op">:</span> <span class="cn">NaN</span>)</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-display" data-execution_count="1">
<pre><code>hₗ (generic function with 1 method)</code></pre>
</div>
</div>
<div id="24" class="cell" data-execution_count="1">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb10"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="fu">gr</span>()</span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-display" data-execution_count="1">
<pre><code>Plots.GRBackend()</code></pre>
</div>
</div>
<div id="26" class="cell" data-hold="true" data-execution_count="1">
<div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb12"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> ys <span class="op">=</span> <span class="fu">range</span>(<span class="op">-</span><span class="fl">1</span>,<span class="fl">1</span>, length<span class="op">=</span><span class="fl">100</span>)</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a>plt <span class="op">=</span> <span class="fu">surface</span>(xs, ys, hₗ)</span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a>ts <span class="op">=</span> cpsₗ <span class="co"># 2pi/3 and 4pi/3 by above</span></span>
<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a>xs, ys <span class="op">=</span> <span class="fu">cos</span>.(ts), <span class="fu">sin</span>.(ts)</span>
<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a>zs <span class="op">=</span> <span class="fu">fₗ</span>.(xs, ys)</span>
<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a><span class="fu">scatter3d!</span>(xs, ys, zs)</span>
<span id="cb12-8"><a href="#cb12-8" aria-hidden="true" tabindex="-1"></a><span class="co">#tuple.(xs, ys, zs)</span></span>
<span id="cb12-9"><a href="#cb12-9" aria-hidden="true" tabindex="-1"></a><span class="co">#plot!(plt, tuple.(xs, ys, zs); linetype=:scatter)</span></span>
<span id="cb12-10"><a href="#cb12-10" aria-hidden="true" tabindex="-1"></a><span class="co">#plt</span></span></code></pre></div><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></div>
<div class="cell-output cell-output-display" data-execution_count="1">
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truncated
</div>
</div>
<p>A contour plot also shows that some—and only one—extrema happens on the interior:</p>
<!--
```lqoxmri
#| hold: true
xs = ys = range(-1,1, length=100)
contour(xs, ys, hₗ)
```
The extrema are identified by the enclosing regions, in this case the one around the point $(1/2, 0)$.
-->
</section>
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quarto/differentiable_vector_calculus/test.jl
+827
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@show 4
using QuizQuestions
using LaTeXStrings
using CalculusWithJulia
using Plots
plotly()
using SymPy
using Roots
@show 6
import Contour: contours, levels, level, lines, coordinates
@show 15
@syms f_x f_y
n = [1, 0, f_x] × [0, 1, f_y]
@show 27
#| hold: true
f(x,y) = 6 - x^2 -y^2
f(x)= f(x...)
a,b = 1, -1/2
# draw surface
xr = 7/4
xs = ys = range(-xr, xr, length=100)
surface(xs, ys, f, legend=false)
# visualize tangent plane as 3d polygon
pt = [a,b]
tplane(x) = f(pt) + gradient(f)(pt) ⋅ (x - [a,b])
pts = [[a-1,b-1], [a+1, b-1], [a+1, b+1], [a-1, b+1], [a-1, b-1]]
plot!(unzip([[pt..., tplane(pt)] for pt in pts])...)
# plot paths in x and y direction through (a,b)
γ_x(t) = pt + t*[1,0]
γ_y(t) = pt + t*[0,1]
plot_parametric!((-xr-a)..(xr-a), t -> [γ_x(t)..., (f∘γ_x)(t)], linewidth=3)
plot_parametric!((-xr-b)..(xr-b), t -> [γ_y(t)..., (f∘γ_y)(t)], linewidth=3)
# draw directional derivatives in 3d and normal
pt = [a, b, f(a,b)]
fx, fy = gradient(f)(a,b)
arrow!(pt, [1, 0, fx], linewidth=3)
arrow!(pt, [0, 1, fy], linewidth=3)
arrow!(pt, [-fx, -fy, 1], linewidth=3) # normal
# draw point in base, x-y, plane
pt = [a, b, 0]
scatter!(unzip([pt])...)
arrow!(pt, [1,0,0], linestyle=:dash)
arrow!(pt, [0,1,0], linestyle=:dash)
@show 33
function tangent_plane_1st_crack(f, pt)
fx, fy = ForwardDiff.gradient(f, pt)
x -> f(x...) + fx * (x[1]-pt[1]) + fy * (x[2]-pt[2])
end
@show 35
function tangent_plane(f, pt)
∇f = ForwardDiff.gradient(f, pt) # using a variable ∇f
x -> f(pt) + ∇f ⋅ (x - pt)
end
@show 46
@syms x, y
@show 47
#| hold: true
f(x,y) = sin(x) * cos(x-y)
f(x) = f(x...)
vars = [x, y]
gradf = diff.(f(x,y), vars) # or use gradient(f, vars) or ∇((f,vars))
pt = [PI/4, PI/3]
gradfa = subs.(gradf, x=>pt[1], y=>pt[2])
f(pt) + gradfa ⋅ (vars - pt)
@show 55
#| hold: true
a = 1
gamma(t) = a * [1 + cos(t), sin(t), 2sin(t/2) ]
P = gamma(1/2)
n1(x,y,z)= [2*(x-a), 2y, 0]
n2(x,y,z) = [2x,2y,2z]
n1(x) = n1(x...)
n2(x) = n2(x...)
t = 1/2
(n1(gamma(t)) × n2(gamma(t))) × gamma'(t)
@show 60
#| hold: true
a, b = 1, 3
f(x,y,z) = (x^2 + ((1+b) * y)^2 + z^2 - 1)^3 - x^2 * z^3 - a * y^2 * z^3
CalculusWithJulia.plot_implicit_surface(f, xlim=-2..2, ylim=-1..1, zlim=-1..2)
@show 71
V(r, h) = pi * r^2 * h
V(v) = V(v...)
a₁ = [1,2]
dx₁ = [0.01, 0.01]
ForwardDiff.gradient(V, a₁) ⋅ dx₁ # or use ∇(V)(a)
@show 73
V(a₁ + dx₁) - V(a₁)
@show 85
#| hold: true
f(x,y,z) = x^4 -x^3 + y^2 + z^2
f(v) = f(v...)
a, b,c = ∇(f)(2,2,2)
"$a x + $b y + $c z = $([a,b,c] ⋅ [2,2,2])"
#@show 92
#| hold: true
@syms a b c d u v
M = [a b; c d]
B = [u, v]
M \ B .|> simplify
@show 96
#| hold: true
#| echo: false
f(x,y) = 2 - x^2 - y^2
g(x,y) = 3 - 2x^2 - (1/3)y^2
xs = ys = range(-3, stop=3, length=100)
zfs = [f(x,y) for x in xs, y in ys]
zgs = [g(x,y) for x in xs, y in ys]
ps = Any[]
pf = surface(xs, ys, f, alpha=0.5, legend=false)
for cl in levels(contours(xs, ys, zfs, [0.0]))
for line in lines(cl)
_xs, _ys = coordinates(line)
plot!(pf, _xs, _ys, 0*_xs, linewidth=3, color=:blue)
end
end
pg = surface(xs, ys, g, alpha=0.5, legend=false)
for cl in levels(contours(xs, ys, zgs, [0.0]))
for line in lines(cl)
_xs, _ys = coordinates(line)
plot!(pg, _xs, _ys, 0*_xs, linewidth=3, color=:red)
end
end
pcnt = plot(legend=false)
for cl in levels(contours(xs, ys, zfs, [0.0]))
for line in lines(cl)
_xs, _ys = coordinates(line)
plot!(pcnt, _xs, _ys, linewidth=3, color=:blue)
end
end
for cl in levels(contours(xs, ys, zgs, [0.0]))
for line in lines(cl)
_xs, _ys = coordinates(line)
plot!(pcnt, _xs, _ys, linewidth=3, color=:red)
end
end
l = @layout([a b c])
plot(pf, pg, pcnt, layout=l)
@show 106
function newton_step(f, g, xn)
M = [ForwardDiff.gradient(f, xn)'; ForwardDiff.gradient(g, xn)']
b = -[f(xn), g(xn)]
Delta = M \ b
xn + Delta
end
@show 108
𝒇(x,y) = 2 - x^2 - y^2
𝒈(x,y) = 3 - 2x^2 - (1/3)y^2
𝒇(v) = 𝒇(v...); 𝒈(v) = 𝒈(v...)
𝒙₀ = [1,1]
𝒙₁ = newton_step(𝒇, 𝒈, 𝒙₀)
@show 110
𝒇(𝒙₁), 𝒈(𝒙₁)
@show 112
𝒙₂ = newton_step(𝒇, 𝒈, 𝒙₁)
𝒙₃ = newton_step(𝒇, 𝒈, 𝒙₂)
𝒙₄ = newton_step(𝒇, 𝒈, 𝒙₃)
𝒙₅ = newton_step(𝒇, 𝒈, 𝒙₄)
𝒙₅, 𝒇(𝒙₅), 𝒈(𝒙₅)
@show 116
function nm(f, g, x, n=5)
for i in 1:n
x = newton_step(f, g, x)
end
x
end
@show 123
#| hold: true
c = 1/2
f(x,y) = 1 - y^2 - c^2
g(x,y) = (1 - x^2) - c^2
f(v) = f(v...); g(v) = g(v...)
nm(f, g, [1/2, 1/3])
@show 148
#| hold: true
@syms x, y, Z()
∂x = solve(diff(x^4 -x^3 + y^2 + Z(x,y)^2, x), diff(Z(x,y),x))
∂y = solve(diff(x^4 -x^3 + y^2 + Z(x,y)^2, y), diff(Z(x,y),y))
∂x, ∂y
@show 158
f(x, p) = cos(x) - p*x
p = 2
xᵅ = find_zero(f, (0, pi/2), p)
@show 160
p = 2
xᵅ = find_zero(f, (0, pi/2), p)
fₓ = ForwardDiff.derivative(x -> f(x,p), xᵅ)
fₚ = ForwardDiff.derivative(p -> f(xᵅ, p), p)
- fₚ / fₓ
@show 163
function find_zero_derivative(f, x₀, p)
xᵅ = find_zero(f, x₀, p)
fₓ = ForwardDiff.derivative(x -> f(x,p), xᵅ)
fₚ = ForwardDiff.derivative(p -> f(xᵅ, p), p)
- fₚ / fₓ
end
F(p) = find_zero_derivative(f, (0, pi/2), p)
plot(F, 0.01, 5) # p > 0
@show 183
#| hold: true
f(x,y)= exp(-(x^2 + y^2)/5) * cos(x^2 + y^2)
xs = ys = range(-4, 4, length=100)
surface(xs, ys, f, legend=false)
@show 190
#| hold: true
f(x,y) = x*y
xs = ys = range(-3, 3, length=100)
surface(xs, ys, f, legend=false)
plot_parametric!(-4..4, t -> [t, 0, f(t, 0)], linewidth=5)
plot_parametric!(-4..4, t -> [0, t, f(0, t)], linewidth=5)
@show 203
fₖ(x,y) = exp(-(x^2 + y^2)/5) * cos(x^2 + y^2)
Hₖ = sympy.hessian(fₖ(x,y), (x,y))
@show 205
H₀₀ = subs.(Hₖ, x=>0, y=>0)
@show 207
H₀₀[1,1] < 0 && det(H₀₀) > 0
@show 209
#| hold: true
gradfₖ = diff.(fₖ(x,y), [x,y])
a = [sqrt(2PI + atan(-Sym(1)//5)), 0]
subs.(gradfₖ, x => a[1], y => a[2])
@show 211
#| hold: true
a = [sqrt(PI + atan(-Sym(1)//5)), 0]
H_a = subs.(Hₖ, x => a[1], y => a[2])
det(H_a)
@show 216
fⱼ(x,y) = 4x*y - x^4 - y^4
gradfⱼ = diff.(fⱼ(x,y), [x,y])
@show 217
all_ptsⱼ = solve(gradfⱼ, [x,y])
ptsⱼ = filter(u -> all(isreal.(u)), all_ptsⱼ)
@show 219
Hⱼ = sympy.hessian(fⱼ(x,y), (x,y))
function classify(H, pt)
Ha = subs.(H, x => pt[1], y => pt[2])
(det=det(Ha), f_xx=Ha[1,1])
end
[classify(Hⱼ, pt) for pt in ptsⱼ]
@show 221
#| hold: true
xs = ys = range(-3/2, 3/2, length=100)
p = surface(xs, ys, fⱼ, legend=false)
for pt ∈ ptsⱼ
scatter!(p, unzip([N.([pt...,fⱼ(pt...)])])...,
markercolor=:black, markersize=5) # add each pt on surface
end
p
@show 228
fₗ(x,y) = x^2 + 2y^2 - x
fₗ(v) = fₗ(v...)
gammaₗ(t) = [cos(t), sin(t)] # traces out x^2 + y^2 = 1 over [0, 2pi]
gₗ = fₗ ∘ gammaₗ
cpsₗ = find_zeros(gₗ', 0, 2pi) # critical points of g
append!(cpsₗ, [0, 2pi])
unique!(cpsₗ)
gₗ.(cpsₗ)
@show 230
inds = [2,4]
cpsₗ[inds]
@show 232
cpsₗ[inds]/pi
@show 234
hₗ(x,y) = fₗ(x,y) * (x^2 + y^2 <= 1 ? 1 : NaN)
@show 235
#| hold: true
xs = ys = range(-1,1, length=100)
surface(xs, ys, hₗ)
ts = cpsₗ # 2pi/3 and 4pi/3 by above
xs, ys = cos.(ts), sin.(ts)
zs = fₗ.(xs, ys)
scatter3d!(xs, ys, zs)
@show 237
#| hold: true
xs = ys = range(-1,1, length=100)
contour(xs, ys, hₗ)
@show 243
@syms x1 y1 x2 y2 x3 y3
d2(p,x) = (p[1] - x[1])^2 + (p[2]-x[2])^2
d2_1, d2_2, d2_3 = d2((x,y), (x1, y1)), d2((x,y), (x2, y2)), d2((x,y), (x3, y3))
exₛ = d2_1 + d2_2 + d2_3
@show 245
gradfₛ = diff.(exₛ, [x,y])
xstarₛ = solve(gradfₛ, [x,y])
@show 248
Hₛ = subs.(hessian(exₛ, [x,y]), x=>xstarₛ[x], y=>xstarₛ[y])
@show 259
usₛ = [[cos(t), sin(t)] for t in (0, 2pi/3, 4pi/3)]
polygon(ps) = unzip(vcat(ps, ps[1:1])) # easier way to plot a polygon
pₛ = scatter([0],[0], markersize=2, legend=false, aspect_ratio=:equal)
asₛ = (1,2,3)
plot!(polygon([a*u for (a,u) in zip(asₛ, usₛ)])...)
[arrow!([0,0], a*u, alpha=0.5) for (a,u) in zip(asₛ, usₛ)]
pₛ
@show 261
asₛ₁ = (1, -1, 3)
scatter([0],[0], markersize=2, legend=false)
psₛₗ = [a*u for (a,u) in zip(asₛ₁, usₛ)]
plot!(polygon(psₛₗ)...)
@show 263
euclid_dist(x; ps=psₛₗ) = sum(norm(x-p) for p in ps)
euclid_dist(x,y; ps=psₛₗ) = euclid_dist([x,y]; ps=ps)
@show 264
#| hold: true
xs = range(-1.5, 1.5, length=100)
ys = range(-3, 1.0, length=100)
p = plot(polygon(psₛₗ)..., linewidth=3, legend=false)
scatter!(p, unzip(psₛₗ)..., markersize=3)
contour!(p, xs, ys, euclid_dist)
# add some gradients along boundary
li(t, p1, p2) = p1 + t*(p2-p1) # t in [0,1]
for t in range(1/100, 1/2, length=3)
pt = li(t, psₛₗ[2], psₛₗ[3])
arrow!(pt, ForwardDiff.gradient(euclid_dist, pt))
pt = li(t, psₛₗ[2], psₛₗ[1])
arrow!(pt, ForwardDiff.gradient(euclid_dist, pt))
end
p
@show 266
#| hold : true
li(t, p1, p2) = p1 + t*(p2-p1)
p = plot(legend=false)
for i in 1:2, j in (i+1):3
plot!(p, t -> euclid_dist(li(t, psₛₗ[i], psₛₗ[j]); ps=psₛₗ), 0, 1)
end
p
@show 280
@syms xₗₛ[1:3] yₗₛ[1:3] α β
li(x, alpha, beta) = alpha + beta * x
d₂(alpha, beta) = sum((y - li(x, alpha, beta))^2 for (y,x) in zip(yₗₛ, xₗₛ))
d₂(α, β)
@show 282
grad_d₂ = diff.(d₂(α, β), [α, β])
@show 283
outₗₛ = solve(grad_d₂, [α, β])
@show 285
subs(outₗₛ[β], sum(xₗₛ) => 0)
@show 292
[k => subs(v, xₗₛ[1]=>1, yₗₛ[1]=>1, xₗₛ[2]=>2, yₗₛ[2]=>3,
xₗₛ[3]=>5, yₗₛ[3]=>8) for (k,v) in outₗₛ]
@show 302
f₂(x,y) = -exp(-((x-1)^2 + 2(y-1/2)^2))
f₂(x) = f₂(x...)
xs₂ = [[0.0, 0.0]] # we store a vector
gammas₂ = [1.0]
for n in 1:5
xn = xs₂[end]
gamma₀ = gammas₂[end]
xn1 = xn - gamma₀ * gradient(f₂)(xn)
dx, dy = xn1 - xn, gradient(f₂)(xn1) - gradient(f₂)(xn)
gamman1 = abs( (dx ⋅ dy) / (dy ⋅ dy) )
push!(xs₂, xn1)
push!(gammas₂, gamman1)
end
[(x, f₂(x)) for x in xs₂]
@show 304
#| hold: true
function surface_contour(xs, ys, f; offset=0)
p = surface(xs, ys, f, legend=false, fillalpha=0.5)
## we add to the graphic p, then plot
zs = [f(x,y) for x in xs, y in ys] # reverse order for use with Contour package
for cl in levels(contours(xs, ys, zs))
lvl = level(cl) # the z-value of this contour level
for line in lines(cl)
_xs, _ys = coordinates(line) # coordinates of this line segment
_zs = offset * _xs
plot!(p, _xs, _ys, _zs, alpha=0.5) # add curve on x-y plane
end
end
p
end
offset = 0
us = vs = range(-1, 2, length=100)
surface_contour(us, vs, f₂, offset=offset)
pts = [[pt..., offset] for pt in xs₂]
scatter3d!(unzip(pts)...)
plot!(unzip(pts)..., linewidth=3)
@show 314
function peaks(x, y)
z = 3 * (1 - x)^2 * exp(-x^2 - (y + 1)^2)
z += -10 * (x / 5 - x^3 - y^5) * exp(-x^2 - y^2)
z += -1/3 * exp(-(x+1)^2 - y^2)
return z
end
peaks(v) = peaks(v...)
@show 315
#| hold: true
xs = range(-3, stop=3, length=100)
ys = range(-2, stop=2, length=100)
Ps = surface(xs, ys, peaks, legend=false)
Pc = contour(xs, ys, peaks, legend=false)
plot(Ps, Pc, layout=2) # combine plots
@show 319
function newton_stepₚ(f, x)
M = ForwardDiff.hessian(f, x)
b = ForwardDiff.gradient(f, x)
x - M \ b
end
@show 321
xₚ = [0, 1.5]
xₚ = newton_stepₚ(peaks, xₚ)
xₚ = newton_stepₚ(peaks, xₚ)
xₚ = newton_stepₚ(peaks, xₚ)
xₚ, ForwardDiff.gradient(peaks, xₚ)
@show 323
Hₚ = ForwardDiff.hessian(peaks, xₚ)
@show 325
#| hold: true
fxx = Hₚ[1,1]
d = det(Hₚ)
fxx, d
@show 335
#| hold: true
g(x,y) = x^2 + 2y^2 -1
g(v) = g(v...)
xs = range(-3, 3, length=100)
ys = range(-1, 4, length=100)
p = plot(aspect_ratio=:equal, legend=false)
contour!(xs, ys, g, levels=[0])
gi(x) = sqrt(1/2*(1-x^2)) # solve for y in terms of x
pts = [[x, gi(x)] for x in (-3/4, -1/4, 1/4, 3/4)]
for pt in pts
arrow!(pt, ForwardDiff.gradient(g, pt) )
end
p
@show 338
#| hold: true
#| echo: false
r(t) = [cos(t), sin(t)/2]
plot_parametric(pi/12..pi/3, r, legend=false, aspect_ratio=true, linewidth=3)
T(t) = -r'(t) / norm(r'(t))
No(t) = T'(t) / norm(T'(t))
t = pi/4
lambda=1/10
scatter!(unzip([r(t)])...)
arrow!(r(t), T(t)*lambda)
arrow!(r(t), No(t)* lambda)
f(x,y)= x^2 + y^2
f(v) = f(v...)
arrow!(r(t), lambda*ForwardDiff.gradient(f, r(t)))
xs = range(0.5,1, length=100)
ys = range(0.1, 0.5, length=100)
contour!(xs, ys, f)
@show 344
#| hold: true
#| echo: false
r(t) = [cos(t), sin(t)/2]
plot_parametric(-pi/6..pi/6,r, legend=false, aspect_ratio=true, linewidth=3)
T(t) = -r'(t) / norm(r'(t))
No(t) = T'(t) / norm(T'(t))
t = 0
lambda=1/10
scatter!(unzip([r(t)])...)
arrow!(r(t), T(t)*lambda)
arrow!(r(t), No(t)* lambda)
f(x,y)= x^2 + y^2
f(v) = f(v...)
arrow!(r(t), lambda*ForwardDiff.gradient(f, r(t)))
xs = range(0.5,1.5, length=100)
ys = range(-0.5, 0.5, length=100)
contour!(xs, ys, f, levels = [.7, .85, 1, 1.15, 1.3])
@show 381
@syms lambda
fₗₐ(x, y) = x^2 - y^2
gₗₐ(x, y) = x^2 + y^2
Lₗₐ(x, y, lambda) = fₗₐ(x,y) - lambda * (gₗₐ(x,y) - 1)
dsₗₐ = solve(diff.(Lₗₐ(x, y, lambda), [x, y, lambda]))
@show 383
[fₗₐ(d[x], d[y]) for d in dsₗₐ]
@show 432
#| hold: true
@syms y y′ λ C
ex = Eq(-λ*y′^2/sqrt(1 + y′^2) + λ*sqrt(1 + y′^2), y - C)
Δ = sqrt(1 + y′^2) / (y - C)
ex1 = Eq(simplify(ex.lhs()*Δ), simplify(ex.rhs() * Δ))
ex2 = Eq(ex1.lhs()^2 - 1, simplify(ex1.rhs()^2) - 1)
@show 457
@syms z lambda1 lambda2
g1(x, y, z) = x^2 + y^2 - z^2
g2(x, y, z) = x - 2z - 3
fₘ(x,y,z)= x^2 + y^2 + z^2
Lₘ(x,y,z,lambda1, lambda2) = fₘ(x,y,z) - lambda1*(g1(x,y,z) - 0) - lambda2*(g2(x,y,z) - 0)
∇Lₘ = diff.(Lₘ(x,y,z,lambda1, lambda2), [x, y, z,lambda1, lambda2])
@show 459
solve(subs.(∇Lₘ, lambda1 .=> 1))
@show 461
outₘ = solve(subs.(∇Lₘ, y .=> 0))
@show 463
[fₘ(d[x], 0, d[z]) for d in outₘ]
@show 498
struct MultiIndex
alpha::Vector{Int}
end
Base.show(io::IO, α::MultiIndex) = println(io, "α = ($(join(α.alpha, ", ")))")
## |α| = α_1 + ... + α_m
Base.length(α::MultiIndex) = sum(α.alpha)
## factorial(α) computes α!
Base.factorial(α::MultiIndex) = prod(factorial(Sym(a)) for a in α.alpha)
## x^α = x_1^α_1 * x_2^α^2 * ... * x_n^α_n
import Base: ^
^(x, α::MultiIndex) = prod(u^a for (u,a) in zip(x, α.alpha))
## ∂^α(ex) = ∂_1^α_1 ∘ ∂_2^α_2 ∘ ... ∘ ∂_n^α_n (ex)
partial(ex::SymPy.SymbolicObject, α::MultiIndex, vars=free_symbols(ex)) = diff(ex, zip(vars, α.alpha)...)
@show 499
@syms w
alpha = MultiIndex([1,2,1,3])
length(alpha) # 1 + 2 + 1 + 3=7
[1,2,3,4]^alpha
exₜ = x^3 * cos(w*y*z)
partial(exₜ, alpha, [w,x,y,z])
@show 501
struct MultiIndices
n::Int
k::Int
end
function Base.length(as::MultiIndices)
n,k = as.n, as.k
n == 1 && return 1
sum(length(MultiIndices(n-1, j)) for j in 0:k) # recursively identify length
end
function Base.iterate(alphas::MultiIndices)
k, n = alphas.k, alphas.n
n == 1 && return ([k],(0, MultiIndices(0,0), nothing))
m = zeros(Int, n)
m[1] = k
betas = MultiIndices(n-1, 0)
stb = iterate(betas)
st = (k, MultiIndices(n-1, 0), stb)
return (m, st)
end
function Base.iterate(alphas::MultiIndices, st)
st == nothing && return nothing
k,n = alphas.k, alphas.n
k == 0 && return nothing
n == 1 && return nothing
# can we iterate the next on
bk, bs, stb = st
if stb==nothing
bk = bk-1
bk < 0 && return nothing
bs = MultiIndices(bs.n, bs.k+1)
val, stb = iterate(bs)
return (vcat(bk,val), (bk, bs, stb))
end
resp = iterate(bs, stb)
if resp == nothing
bk = bk-1
bk < 0 && return nothing
bs = MultiIndices(bs.n, bs.k+1)
val, stb = iterate(bs)
return (vcat(bk, val), (bk, bs, stb))
end
val, stb = resp
return (vcat(bk, val), (bk, bs, stb))
end
@show 503
collect(MultiIndices(2, 3))
@show 505
union((collect(MultiIndices(2, i)) for i in 0:3)...)
@show 507
k = 4
length(MultiIndices(3, k+1))
@show 509
#| hold: true
@syms 𝐅() a[1:3] dx[1:3]
sum(partial(𝐅(a...), α, a) / factorial(α) * dx^α for k in 0:3 for α in MultiIndex.(MultiIndices(3, k))) # 3rd order
@show 513
#| hold: true
#| echo: false
f(x,y) = sqrt(x + y)
f(v) = f(v...)
pt = [2,2]
dxdy = [.1, .2]
val = f(pt) + dot(ForwardDiff.gradient(f, pt), dxdy)
numericq(val)
@show 516
#| hold: true
#| echo: false
f(x,y,z) = x*y + y*z + z*x
f(v) = f(v...)
pt = [1,1,1]
dx = [0.1, 0.0, -0.1]
val = f(pt) + ∇(f)(pt) ⋅ dx
numericq(val)
@show 519
#| hold: true
#| echo: false
f(x,y,z) = x*y + y*z + z*x - 8
f(v) = f(v...)
pt = [1,1,1]
n = ∇(f)(pt)
d = dot(n, pt)
choices = [
raw"`` x + y + z = 3``",
raw"`` 2x + y - 2z = 1``",
raw"`` x + 2y + 3z = 6``"
]
answ = 1
radioq(choices, answ)
@show 523
#| hold: true
#| echo: false
choices = [
raw"`` \langle 2xy + y^2 + y, 2xy + x^2 + x\rangle``",
raw"`` y^2 + y, x^2 + x``",
raw"`` \langle 2y + y^2, 2x + x^2``"
]
answ = 1
radioq(choices, answ)
@show 527
#| hold: true
#| echo: false
yesnoq(true)
@show 529
#| hold: true
#| echo: false
f(x,y) = x*y + x*y^2 + x^2 * y
f(v) = f(v...)
val = det(ForwardDiff.hessian(f, [-1/3, -1/3]))
numericq(val)
@show 531
#| hold: true
#| echo: false
choices = [
L"The function $f$ has a local minimum, as $f_{xx} > 0$ and $d >0$",
L"The function $f$ has a local maximum, as $f_{xx} < 0$ and $d >0$",
L"The function $f$ has a saddle point, as $d < 0$",
L"Nothing can be said, as $d=0$"
]
answ = 2
radioq(choices, answ, keep_order=true)
@show 535
#| hold: true
#| results: "hidden"
f(x,y) = x + 2x^2 + x^3 + y + 2x*y + y^2
@syms x::real y::real
gradf = gradient(f(x,y), [x,y])
@show 536
#| hold: true
#| echo: false
yesnoq(true)
@show 538
#| hold: true
#| results: "hidden"
f(x,y) = x + 2x^2 + x^3 + y + 2x*y + y^2
@syms x::real y::real
gradf = gradient(f(x,y), [x,y])
solve(gradf, [x,y])
@show 539
#| hold: true
#| echo: false
numericq(2)
@show 541
#| hold: true
f(x,y) = x + 2x^2 + x^3 + y + 2x*y + y^2
@syms x::real y::real
gradf = gradient(f(x,y), [x,y])
sympy.hessian(f(x,y), [x,y])
@show 543
#| hold: true
#| echo: false
choices = [
L"The function $f$ has a local minimum, as $f_{xx} > 0$ and $d >0$",
L"The function $f$ has a local maximum, as $f_{xx} < 0$ and $d >0$",
L"The function $f$ has a saddle point, as $d < 0$",
L"Nothing can be said, as $d=0$",
L"The test does not apply, as $\nabla{f}$ is not $0$ at this point."
]
answ = 3
radioq(choices, answ, keep_order=true)
@show 545
#| hold: true
#| echo: false
choices = [
L"The function $f$ has a local minimum, as $f_{xx} > 0$ and $d >0$",
L"The function $f$ has a local maximum, as $f_{xx} < 0$ and $d >0$",
L"The function $f$ has a saddle point, as $d < 0$",
L"Nothing can be said, as $d=0$",
L"The test does not apply, as $\nabla{f}$ is not $0$ at this point."
]
answ = 1
radioq(choices, answ, keep_order=true)
@show 547
#| hold: true
#| echo: false
choices = [
L"The function $f$ has a local minimum, as $f_{xx} > 0$ and $d >0$",
L"The function $f$ has a local maximum, as $f_{xx} < 0$ and $d >0$",
L"The function $f$ has a saddle point, as $d < 0$",
L"Nothing can be said, as $d=0$",
L"The test does not apply, as $\nabla{f}$ is not $0$ at this point."
]
answ = 5
radioq(choices, answ, keep_order=true)
@show 553
#| hold: true
#| echo: false
yesnoq(true)
@show 557
#| hold: true
#| echo: false
yesnoq(false)
@show 559
#| hold: true
#| echo: false
choices =[
"It is the determinant of the Hessian",
L"It isn't, $b^2-4ac$ is from the quadratic formula"
]
answ = 1
radioq(choices, answ)
@show 561
#| hold: true
#| echo: false
choices = [
L"That $a>0$ and $4ac-b^2 > 0$",
L"That $a<0$ and $4ac-b^2 > 0$",
L"That $4ac-b^2 < 0$"
]
answ = 2
radioq(choices, answ, keep_order=true)
@show 563
#| hold: true
#| echo: false
choices = [
L"That $a>0$ and $4ac-b^2 > 0$",
L"That $a<0$ and $4ac-b^2 > 0$",
L"That $4ac-b^2 < 0$"
]
answ = 3
radioq(choices, answ, keep_order=true)
@show 569
#| hold: true
#| echo: false
yesnoq(true)
@show 571
#| echo: false
choices = [
raw"`` \langle 2x, 2y\rangle``",
raw"`` \langle 2x, y^2\rangle``",
raw"`` \langle x^2, 2y \rangle``"
]
answ = 1
radioq(choices, answ)
@show 573
f(x,y) = exp(-x^2-y^2) * (2x^2 + y^2)
f(v) = f(v...)
r(t) = sqrt(3)*[cos(t), sin(t)]
rat(x) = abs(x[1]/x[2]) - 1
fn = rat ∘ ∇(f) ∘ r
ts = fzeros(fn, 0, 2pi)
@show 575
#| eval: false
#| echo: false
f(x,y) = exp(-x^2-y^2) * (2x^2 + y^2)
r(t) = sqrt(3)*[cos(t), sin(t)]
rat(x) = abs(x[1]/x[2]) - 1
fn = rat ∘ ∇(splat(f)) ∘ r
ts = fzeros(fn, 0, 2pi)
val = maximum((splat(u)∘r).(ts))
numericq(val)
quarto/differentiable_vector_calculus/test.qmd
+98
View File
@@ -0,0 +1,98 @@
# Applications with scalar functions
{{< include ../_common_code.qmd >}}
This section uses these add-on packages:
```{julia}
using CalculusWithJulia
using Plots
plotly()
using SymPy
using Roots
```
##### Example
Consider the function $f(x,y) = x^2 + 3y^2 -x$ over the region $x^2 + y^2 \leq 1$. This is a continuous function over a closed set, so will have both an absolute maximum and minimum. Find these from an investigation of the critical points and the boundary points.
The gradient is easily found: $\nabla{f} = \langle 2x - 1, 6y \rangle$, and is $\vec{0}$ only at $\vec{a} = \langle 1/2, 0 \rangle$. The Hessian is:
$$
H =
\begin{bmatrix}
2 & 0\\
0 & 6
\end{bmatrix}.
$$
At $\vec{a}$ this has positive determinant and $f_{xx} > 0$, so $\vec{a}$ corresponds to a *local* minimum with values $f(\vec{a}) = (1/2)^2 + 3(0) - 1/2 = -1/4$. The absolute maximum and minimum may occur here (well, not the maximum) or on the boundary, so that must be considered. In this case we can easily parameterize the boundary and turn this into the univariate case:
```{julia}
fₗ(x,y) = x^2 + 2y^2 - x
gammaₗ(t) = [cos(t), sin(t)] # traces out x^2 + y^2 = 1 over [0, 2pi]
gₗ = splat(fₗ) ∘ gammaₗ
cpsₗ = find_zeros(gₗ', 0, 2pi) # critical points of g
append!(cpsₗ, [0, 2pi])
unique!(cpsₗ)
gₗ.(cpsₗ)
```
We see that maximum value is `2.25` and that the interior point, $\vec{a}$, will be where the minimum value occurs. To see exactly where the maximum occurs, we look at the values of gamma:
```{julia}
inds = [2,4]
cpsₗ[inds]
```
These are multiples of $\pi$:
```{julia}
cpsₗ[inds]/pi
```
So we have the maximum occurs at the angles $2\pi/3$ and $4\pi/3$. Here we visualize, using a hacky trick of assigning `NaN` values to the function to avoid plotting outside the circle:
```{julia}
hₗ(x,y) = fₗ(x,y) * (x^2 + y^2 <= 1 ? 1 : NaN)
```
```{julia}
gr()
```
```{julia}
#| hold: true
xs = ys = range(-1,1, length=100)
plt = surface(xs, ys, hₗ)
ts = cpsₗ # 2pi/3 and 4pi/3 by above
xs, ys = cos.(ts), sin.(ts)
zs = fₗ.(xs, ys)
scatter3d!(xs, ys, zs)
#tuple.(xs, ys, zs)
#plot!(plt, tuple.(xs, ys, zs); linetype=:scatter)
#plt
```
A contour plot also shows that some---and only one---extrema happens on the interior:
<!--
```{julia}
#| hold: true
xs = ys = range(-1,1, length=100)
contour(xs, ys, hₗ)
```
The extrema are identified by the enclosing regions, in this case the one around the point $(1/2, 0)$.
-->
quarto/differentiable_vector_calculus/vector_fields.qmd
+2 -2
View File
@@ -9,7 +9,7 @@ This section uses these add-on packages:
```{julia}
using CalculusWithJulia
using Plots
plotly()
#plotly()
using SymPy
using ForwardDiff
using LinearAlgebra
@@ -454,7 +454,7 @@ surface(unzip(surf.(ts, θs'))...; legend=false)
```{julia}
#| echo: false
plotly()
#plotly()
nothing
```