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quarto/ODEs/differential_equations.qmd
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@@ -260,7 +260,7 @@ This very clearly shows the sharp dependence on the value of $b$; below some lev
The function `recovered` is of two variables returning a single value. In subsequent sections we will see a few $3$-dimensional plots that are common for such functions, here we skip ahead and show how to visualize multiple function plots at once using "`z`" values in a graph. The function `recovered` is of two variables returning a single value. In subsequent sections we will see a few $3$-dimensional plots that are common for such functions, here we skip ahead and show how to visualize multiple function plots at once using "`z`" values in a graph.
::: {#fig-recoverd-over-various-k-values} ::: {#fig-recovered-over-various-k-values}
```{julia} ```{julia}
#| hold: true #| hold: true
k, ks = 0.1, 0.2:0.1:0.9 # first `k` and then the rest k, ks = 0.1, 0.2:0.1:0.9 # first `k` and then the rest
quarto/ODEs/euler.qmd
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@@ -184,7 +184,7 @@ for i in 1:n
end end
``` ```
So how did we do? @fig-euler-yp-yx-n-5 shows the graph of the exact answer and the stiched-together answer. So how did we do? @fig-euler-yp-yx-n-5 shows the graph of the exact answer and the stitched-together answer.
::: {#fig-euler-yp-yx-n-5} ::: {#fig-euler-yp-yx-n-5}
quarto/alternatives/giac.qmd
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@@ -9,7 +9,7 @@ format:
# Using the Giac symbolic math engine within Julia # Using the Giac symbolic math engine within Julia
This page describes the use of symbolic math for some topics of Calculus within `Julia` utilizing the [Giac](http://www-fourier.ujf-grenoble.fr/~parisse/giac.html) library. Giac is accessed through `Giac.jl`. Giac is a computer algebra system (CAS) implemented in C++; `Giac.jl` is an interface. See this [pdf documentation](https://www-fourier.univ-grenoble-alpes.fr/~parisse/giac/cascmd_en.pdf) for a comprehensive set of topics or the `Julia` documention for the different commands discussed. This page describes the use of symbolic math for some topics of Calculus within `Julia` utilizing the [Giac](http://www-fourier.ujf-grenoble.fr/~parisse/giac.html) library. Giac is accessed through `Giac.jl`. Giac is a computer algebra system (CAS) implemented in C++; `Giac.jl` is an interface. See this [pdf documentation](https://www-fourier.univ-grenoble-alpes.fr/~parisse/giac/cascmd_en.pdf) for a comprehensive set of topics or the `Julia` documentation for the different commands discussed.
There are other possible choices for symbolic math within the Julia ecosystem: There are other possible choices for symbolic math within the Julia ecosystem:
@@ -1680,7 +1680,7 @@ This new object can be manipulated as any other symbolic expression.
## Parametric description of functions ## Parametric description of functions
Vectors can be used to describe curves parameterically (where the $x$ and $y$ motions are modeled by a third variable, typically $t$ for time). We describe this in a bit more detail later when discussing univariate, vector-valued functions. Vectors can be used to describe curves parametrically (where the $x$ and $y$ motions are modeled by a third variable, typically $t$ for time). We describe this in a bit more detail later when discussing univariate, vector-valued functions.
Consider a person on an advanced ferris wheel with coordinates $x(t)$ and $y(t)$ describing position. The wheel of radius $r$ is centered $R$ units above and circles at radius $\omega$, the car has radius $r_0$ and rotates $6$ times faster. Consider a person on an advanced ferris wheel with coordinates $x(t)$ and $y(t)$ describing position. The wheel of radius $r$ is centered $R$ units above and circles at radius $\omega$, the car has radius $r_0$ and rotates $6$ times faster.
@@ -2176,7 +2176,7 @@ integrate.(spiral, t, 0, pi)
##### Line integrals ##### Line integrals
A line integral might be generically written over a curve $C$ or with a paramterization, $r(t)$, of $C$, yielding A line integral might be generically written over a curve $C$ or with a parameterization, $r(t)$, of $C$, yielding
$$ $$
I = \int_C f(\vec{x}) ds = \int_a^b f(r(t)) dt I = \int_C f(\vec{x}) ds = \int_a^b f(r(t)) dt
@@ -2379,8 +2379,8 @@ For example, to integrate $F(x,y) = x^2 \cdot y^3$ over the triangular region fo
```{julia} ```{julia}
@giac_var x y @giac_var x y
F = x^2 * y^3 F = x^2 * y^3
Iy = integrate(F, y, 0, 1-x) # integrating int_{x=0}^1 int_{y=0}^{1-x} F(x,y) dy dx I_y = integrate(F, y, 0, 1-x) # integrating int_{x=0}^1 int_{y=0}^{1-x} F(x,y) dy dx
integrate(Iy, x, 0, 1) integrate(I_y, x, 0, 1)
``` ```
(Integrating is made easy, but *not* the task of identifying valid endpoints to describe the region integrated over.) (Integrating is made easy, but *not* the task of identifying valid endpoints to describe the region integrated over.)
quarto/alternatives/makie_plotting.qmd
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@@ -510,7 +510,7 @@ arrows3d!(Point3.(us), Point3.(dus))
current_figure() current_figure()
``` ```
Plot of tangent lines in both two and three dimenstions Plot of tangent lines in both two and three dimensions
::: :::
@@ -851,9 +851,9 @@ The manual construction of a figure and an axis object will be further discussed
### Three dimensional contour plots ### Three dimensional contour plots
The `contour` function can also plot $3$-dimensional contour plots. Concentric spheres, contours of $x^2 + y^2 + z^2 = c$ for $c > 0$ are presented in @fig-makie-countour-three-d. The `contour` function can also plot $3$-dimensional contour plots. Concentric spheres, contours of $x^2 + y^2 + z^2 = c$ for $c > 0$ are presented in @fig-makie-contour-three-d.
::: {#fig-makie-countour-three-d} ::: {#fig-makie-contour-three-d}
```{julia} ```{julia}
f(x,y,z) = x^2 + y^2 + z^2 f(x,y,z) = x^2 + y^2 + z^2
xs = ys = zs = range(-3, 3, length=100) xs = ys = zs = range(-3, 3, length=100)
quarto/alternatives/symbolics.qmd
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@@ -762,7 +762,7 @@ Symbolics.jacobian(eqs, [x,y])
## Integration ## Integration
The `SymbolicIntegration` package provides two means to integration *univariate* functions using either the Risch alogorithm or a rules-based approach. The `SymbolicIntegration` package provides two means to integration *univariate* functions using either the Risch algorithm or a rules-based approach.
```{julia} ```{julia}
using SymbolicIntegration, Symbolics using SymbolicIntegration, Symbolics
quarto/basics/numbers_types-II.html
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quarto/basics/numbers_types-II.qmd
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@@ -657,7 +657,7 @@ val = sum(2^i for i in 0:10)
numericq(val) numericq(val)
``` ```
The value `1023` is called a bias. The exponent is coded as the binary value as a positve integer minus $1023$. A bias is used, and not the two's complement format, as storage with a bias makes multiplying by powers of $2$ as easy as shifting the bits. The value `1023` is called a bias. The exponent is coded as the binary value as a positive integer minus $1023$. A bias is used, and not the two's complement format, as storage with a bias makes multiplying by powers of $2$ as easy as shifting the bits.
To find the storage for, say, $2^4 + 2^2 + 2^0$ or `00000010101` we would add `1023` or `01111111111` and see: To find the storage for, say, $2^4 + 2^2 + 2^0$ or `00000010101` we would add `1023` or `01111111111` and see:
quarto/basics/numbers_types.qmd
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@@ -677,7 +677,7 @@ val = sum(2^i for i in 0:10)
numericq(val) numericq(val)
``` ```
The value `1023` is called a bias. The exponent is coded as the binary value as a positve integer minus $1023$. A bias is used, and not the two's complement format, as storage with a bias makes multiplying by powers of $2$ as easy as shifting the bits. The value `1023` is called a bias. The exponent is coded as the binary value as a positive integer minus $1023$. A bias is used, and not the two's complement format, as storage with a bias makes multiplying by powers of $2$ as easy as shifting the bits.
To find the storage for, say, $2^4 + 2^2 + 2^0$ or `00000010101` we would add `1023` or `01111111111` and see: To find the storage for, say, $2^4 + 2^2 + 2^0$ or `00000010101` we would add `1023` or `01111111111` and see:
quarto/basics/vectors.qmd
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@@ -33,7 +33,7 @@ $$
This formula agrees with Pythagorean's theorem for right triangles. This formula agrees with Pythagorean's theorem for right triangles.
For $n$-dimensional points, the same formula may be used with adjustements to the notation. Suppose $P = (x_1, x_2, \dots, x_n)$ and $Q = (y_1, y_2, \cdots, y_n)$. then For $n$-dimensional points, the same formula may be used with adjustments to the notation. Suppose $P = (x_1, x_2, \dots, x_n)$ and $Q = (y_1, y_2, \cdots, y_n)$. then
$$ $$
d = \overline{PQ} = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2}. d = \overline{PQ} = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2 + \cdots + (x_n - y_n)^2}.
@@ -56,7 +56,7 @@ A two-dimensional vector has two components, as does a point in the Cartesian pl
Suppose a vector is defined as being between two points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ with $P$ the endpoint, then the vector connecting $P$ to $Q$ would be $\vec{v} = \langle x_2 - x_1, ~ y_2 - y_1 \rangle$. Suppose a vector is defined as being between two points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ with $P$ the endpoint, then the vector connecting $P$ to $Q$ would be $\vec{v} = \langle x_2 - x_1, ~ y_2 - y_1 \rangle$.
When $P = (0,0)$, the the point $Q = (x_2, y_2)$ has the same components as the vector $\vec{v} = \langle x_2, ~ y_2 \rangle$ leading to a natural indentification between a point and vector, though they represent different things. When $P = (0,0)$, the the point $Q = (x_2, y_2)$ has the same components as the vector $\vec{v} = \langle x_2, ~ y_2 \rangle$ leading to a natural identification between a point and vector, though they represent different things.
@@ -767,7 +767,7 @@ Broadcasting is a widely used and powerful surface syntax which we will employ o
#### Mapping a function over a collection #### Mapping a function over a collection
The `map` function is very much related to broadcasting, in that it applies a function to each element of an iterable. When more than one iterable is specifed, `map` applies the function to the `zip`ped iterables. Unlike broadcasting, `map` does not reshape the underlying iterables. The `map` function is very much related to broadcasting, in that it applies a function to each element of an iterable. When more than one iterable is specified, `map` applies the function to the `zip`ped iterables. Unlike broadcasting, `map` does not reshape the underlying iterables.
Similarly named functions are found in many different programming languages, as `map` is one of the foundational higher-order, functional programming operations. (The "dot" broadcast is mostly limited to `Julia` and mirrors a similar usage of a dot in `MATLAB`.) For those familiar with other programming languages, using `map` may seem more natural. Its syntax is `map(f, xs)`. Additional iterables are passed after `xs`. Similarly named functions are found in many different programming languages, as `map` is one of the foundational higher-order, functional programming operations. (The "dot" broadcast is mostly limited to `Julia` and mirrors a similar usage of a dot in `MATLAB`.) For those familiar with other programming languages, using `map` may seem more natural. Its syntax is `map(f, xs)`. Additional iterables are passed after `xs`.
@@ -787,7 +787,7 @@ sum(map(sin, xs))
This has a performance drawback---there are two passes through the container, one to apply `sin` another to add. This has a performance drawback---there are two passes through the container, one to apply `sin` another to add.
For this task, the `sum` reduction, as others, allows a function to be specified that is applied to each value in the container while the sum is being computed. This argument comes first. A recommened alternative to the previous would be: For this task, the `sum` reduction, as others, allows a function to be specified that is applied to each value in the container while the sum is being computed. This argument comes first. A recommended alternative to the previous would be:
```{julia} ```{julia}
sum(sin, xs) sum(sin, xs)
quarto/derivatives/curve_sketching.qmd
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@@ -112,7 +112,7 @@ let
end end
# (2) periodic behaviour, # (2) periodic behaviour,
i == 2 && (title = "No periodic behavious") i == 2 && (title = "No periodic behaviours")
if i >= 2 if i >= 2
end end
quarto/derivatives/first_second_derivatives.qmd
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@@ -58,7 +58,7 @@ A parallel definition with $a < b$ implying $f(a) > f(b)$ would be used for a *s
::: :::
We introduce a helper function `plotif` from the `CalculusWithJulia` package that highlights the graph of a function $f$ when another function $g(x)$ satisifies $g(x) \geq 0$. This function is called as `plotif(f, g, a, b)`. We introduce a helper function `plotif` from the `CalculusWithJulia` package that highlights the graph of a function $f$ when another function $g(x)$ satisfies $g(x) \geq 0$. This function is called as `plotif(f, g, a, b)`.
To see where a function is positive, we simply pass the function object in for *both* `f` and `g` above. For example, in @fig-plotif-sin-sin-minus-2pi-2pi we look at where $f(x) = \sin(x)$ is positive. To see where a function is positive, we simply pass the function object in for *both* `f` and `g` above. For example, in @fig-plotif-sin-sin-minus-2pi-2pi we look at where $f(x) = \sin(x)$ is positive.
quarto/differentiable_vector_calculus/test.html
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<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">
<img 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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 $(1/2, 0)$.
-->
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quarto/differentiable_vector_calculus/test.jl
-827
View File
@@ -1,827 +0,0 @@
@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
@@ -1,98 +0,0 @@
# 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/integrals/area_between_curves.qmd
+1 -1
View File
@@ -718,7 +718,7 @@ let
O = (0, 0) O = (0, 0)
x1, y1 = P = (2, 2) x1, y1 = P = (2, 2)
x2, y2 = Q = (1, 3) x2, y2 = Q = (1, 3)
plt = plot(; legend=false, aspect_ratio=:equal, framestyle=:orgin) plt = plot(; legend=false, aspect_ratio=:equal, framestyle=:origin)
plot!(plt, [O,P,Q,O]; line=(1, :black)) plot!(plt, [O,P,Q,O]; line=(1, :black))
scatter!(plt, [O, P, Q]; marker=(5, :black)) scatter!(plt, [O, P, Q]; marker=(5, :black))
annotate!(plt, [ annotate!(plt, [
quarto/integrals/center_of_mass.qmd
+2 -2
View File
@@ -227,10 +227,10 @@ The bottom integral is just the area (or total mass if the $\rho$ were not cance
##### Example ##### Example
Find the center of mass formed by the intersection of the parabolas $y=1 - x^2$ and $y=(x-1)^2 - 2$. @fig-center-of-mass-of-two-parabola-1-minus-xsquared-and-x-minus-1-sqared-minus-2 shows that for the $x$ direction, it is close to $1/2$. Find the center of mass formed by the intersection of the parabolas $y=1 - x^2$ and $y=(x-1)^2 - 2$. @fig-center-of-mass-of-two-parabola-1-minus-xsquared-and-x-minus-1-squared-minus-2 shows that for the $x$ direction, it is close to $1/2$.
::: {#fig-center-of-mass-of-two-parabola-1-minus-xsquared-and-x-minus-1-sqared-minus-2} ::: {#fig-center-of-mass-of-two-parabola-1-minus-xsquared-and-x-minus-1-squared-minus-2}
```{julia} ```{julia}
#| echo: false #| echo: false
f1(x) = 1 - x^2 f1(x) = 1 - x^2
quarto/integrals/substitution.qmd
+1 -1
View File
@@ -22,7 +22,7 @@ using SymPy
The technique of $u$-[substitution](https://en.wikipedia.org/wiki/Integration_by_substitution) is derived from reversing the chain rule: $[f(g(x))]' = f'(g(x)) g'(x)$. The technique of $u$-[substitution](https://en.wikipedia.org/wiki/Integration_by_substitution) is derived from reversing the chain rule: $[f(g(x))]' = f'(g(x)) g'(x)$.
::: {.definition title="Subsitution"} ::: {.definition title="Substitution"}
Suppose that $g$ is continuous and $u(x)$ is differentiable with $u'(x)$ being Riemann integrable. Then both these integrals are defined and are equal: Suppose that $g$ is continuous and $u(x)$ is differentiable with $u'(x)$ being Riemann integrable. Then both these integrals are defined and are equal:
quarto/integrals/surface_area.qmd
+1 -1
View File
@@ -668,7 +668,7 @@ f(t) = 2(1 + cos(t)) * sin(t)
plot(g, f, 0, 1pi) plot(g, f, 0, 1pi)
``` ```
Paremeterized curve to rotate about $x$ axis Parameterized curve to rotate about $x$ axis
::: :::
The integrand simplifies to $8\sqrt{2}\pi \sin(t) (1 + \cos(t))^{3/2}$. This lends itself to $u$-substitution with $u=\cos(t)$. The integrand simplifies to $8\sqrt{2}\pi \sin(t) (1 + \cos(t))^{3/2}$. This lends itself to $u$-substitution with $u=\cos(t)$.
quarto/precalc/julia_overview.qmd
+1 -1
View File
@@ -626,7 +626,7 @@ xs = range(0, 2pi, length=251)
ys = [sin(2x) + sin(3x) + sin(4x) for x in xs] ys = [sin(2x) + sin(3x) + sin(4x) for x in xs]
plot(xs, ys) plot(xs, ys)
``` ```
Plot of $f(x) = \sin(2x) + \sin(3x) + \sin(4x)$ over $[0, 2\pi]$ made by constructing vectors `xs` , `ys` holding $x$ and $y$ coordiinates of points to include Plot of $f(x) = \sin(2x) + \sin(3x) + \sin(4x)$ over $[0, 2\pi]$ made by constructing vectors `xs` , `ys` holding $x$ and $y$ coordinates of points to include
::: :::
There are different plotting interfaces. Though not shown, all of these `plot` commands produce a plot of `f`, though with minor differences: There are different plotting interfaces. Though not shown, all of these `plot` commands produce a plot of `f`, though with minor differences:
quarto/precalc/transformations.qmd
+1 -1
View File
@@ -89,7 +89,7 @@ Starting with two functions and composing them requires nothing more than a soli
::: {.callout-note} ::: {.callout-note}
## Infix operator ## Infix operator
Composition of two functions does have an infix operator, `∘`, entered as `\circ[tab]`. This mirrors the mathematical usage of this syntax, though the order of operations are such that calling the composed function on a value requires an extra set of parentheses: `(f∘g)(x)`, as the expresssion `f∘g(x)` evaluates `g(x)` before the composition. Composition of two functions does have an infix operator, `∘`, entered as `\circ[tab]`. This mirrors the mathematical usage of this syntax, though the order of operations are such that calling the composed function on a value requires an extra set of parentheses: `(f∘g)(x)`, as the expression `f∘g(x)` evaluates `g(x)` before the composition.
::: :::
quarto/precalc/trig_functions.qmd
+2 -2
View File
@@ -1173,7 +1173,7 @@ The sine function is an *odd* function.
#| echo: false #| echo: false
choices = ["odd", "even", "neither"] choices = ["odd", "even", "neither"]
answer = 1 answer = 1
explanation = "subsitute `-x` into the exponential formula to see" explanation = "substitute `-x` into the exponential formula to see"
buttonq(choices, answer; explanation) buttonq(choices, answer; explanation)
``` ```
@@ -1185,7 +1185,7 @@ buttonq(choices, answer; explanation)
#| echo: false #| echo: false
choices = ["odd", "even", "neither"] choices = ["odd", "even", "neither"]
answer = 2 answer = 2
explanation = L"The value of $\cosh(-x)$ is the $y$ position of the point $(x,y)$ refelected through the $y$ axis, so is unchanged." explanation = L"The value of $\cosh(-x)$ is the $y$ position of the point $(x,y)$ reflected through the $y$ axis, so is unchanged."
buttonq(choices, answer; explanation) buttonq(choices, answer; explanation)
``` ```