From ed5434bb1c24f3ee5a27d1ba1fee382aa906c2e3 Mon Sep 17 00:00:00 2001 From: jverzani Date: Tue, 28 Mar 2023 11:02:18 -0400 Subject: [PATCH] tweaks --- quarto/README-quarto.md | 22 ++++++++ .../derivatives/first_second_derivatives.qmd | 10 ++++ .../plots_plotting.qmd | 27 ++++++++- .../vector_valued_functions.qmd | 1 + quarto/index.qmd | 2 +- .../double_triple_integrals.qmd | 3 + quarto/integrals/area.qmd | 29 +++++----- quarto/integrals/area_between_curves.qmd | 18 +++++- quarto/integrals/figures/excess-deaths.png | Bin 0 -> 145726 bytes quarto/precalc/calculator.qmd | 52 ++++++++++++++---- quarto/precalc/figures/swale.png | Bin 0 -> 10385 bytes quarto/precalc/polynomial.qmd | 7 +++ quarto/precalc/variables.qmd | 48 ++++++++++++++++ 13 files changed, 187 insertions(+), 32 deletions(-) create mode 100644 quarto/integrals/figures/excess-deaths.png create mode 100644 quarto/precalc/figures/swale.png diff --git a/quarto/README-quarto.md b/quarto/README-quarto.md index d158e74..e0bcd02 100644 --- a/quarto/README-quarto.md +++ b/quarto/README-quarto.md @@ -92,6 +92,12 @@ DONE * modify sympy's show method ## TODO +* Old and new astronomy; : Proctor, Richard Anthony, 1837-1888 : Free Download, Borrow, and Streaming : Internet Archive + + +https://archive.org/details/oldnewastronomy00procuoft +https://ia800908.us.archive.org/16/items/oldnewastronomy00procuoft/oldnewastronomy00procuoft.pdf has interesting epicycle figures on p 186/886. Could add these in. + * ImageFile -> ![]() as much as possible * use pandoc references(?) * mermaid, ojs? @@ -103,3 +109,19 @@ DONE * remove pinned header DONE * clean up directory DONE (?) * JSXGraph files WON'T DO * download links to Pluto .jl files (if we have .jmd, but we might deprecate...) For *now* .jmd is derprecated; though we keep the files around .... + +## Cone, general +# https://discourse.julialang.org/t/general-plotting-code-for-cone-in-3d-with-glmakie-or-plots/92104/3 + +basecurve(u) = [cos(u), sin(u) + sin(u/2), 0] +Vertex = [0, 3/4, 3] +function Cone(u, t) + x, y, z = basecurve(u) + t * ([x,y,z] - Vertex) + Vertex +end + +us = range(0, 2pi, length=50) +ts = range(0, 1, length=50) + +xyzs = Cone.(us, ts') +surface(CalculusWithJulia.unzip(xyzs)) diff --git a/quarto/derivatives/first_second_derivatives.qmd b/quarto/derivatives/first_second_derivatives.qmd index 2a3aa56..caebbc4 100644 --- a/quarto/derivatives/first_second_derivatives.qmd +++ b/quarto/derivatives/first_second_derivatives.qmd @@ -668,6 +668,16 @@ answ = 3 radioq(choices, answ) ``` +###### Question + +Consider the function `f(x) = airyai(x)`. What is the value of $f''(x) - x\cdot f(x)$ at $x=\pi$? + +```{julia} +#| echo: false +numericq(0; hint=raw"Any value of $x$ will produce the same answer") +``` + + ###### Question diff --git a/quarto/differentiable_vector_calculus/plots_plotting.qmd b/quarto/differentiable_vector_calculus/plots_plotting.qmd index 0c82d20..3c787d1 100644 --- a/quarto/differentiable_vector_calculus/plots_plotting.qmd +++ b/quarto/differentiable_vector_calculus/plots_plotting.qmd @@ -373,7 +373,7 @@ surface(xs, ys, zs) :::{.callout-note} ## Note The above may not work with all backends for `Plots`, even if those that support 3D graphics. - ::: +::: For convenience, the `plot_parametric` function from `CalculusWithJulia` can produce these plots using interval notation, `a..b`, and a function: @@ -384,6 +384,31 @@ F(theta, phi) = [X(1, theta, phi), Y(1, theta, phi), Z(1, theta, phi)] plot_parametric(0..pi, 0..pi/2, F) ``` +##### Example, the general cone + +The general equation of cone consists of a vertex, $V$, and a base curve. The cone consists of all line segments connecting $V$ to the base curve, here parameterized and in the $x-y$ plane: $r(u) = \langle x(u), y(u), 0 \rangle$. The equations for the cone are: + +$$ +\frac{x - V_x}{x(u)-V_x} = \frac{y - V_y}{y(u)-V_y} = \frac{z - V_z}{z(u)-V_z} = t, +$$ + +where $t \in [0,1]$ This gives a vector-valued function + +$$ +F(u, t) = t * (r(u) - V) + V, \quad \alpha \leq u \leq \beta, 0 \leq t \leq 1. +$$ + +To illustrate, we have: + +```{julia} +# cf. https://discourse.julialang.org/t/general-plotting-code-for-cone-in-3d-with-glmakie-or-plots/92104/3 + +basecurve(u) = [cos(u), sin(u) + sin(u/2), 0] +Vertex = [0, 3/4, 3] +Cone(u, t) = t * (basecurve(u) - Vertex) + Vertex +plot_parametric(0..2pi, 0..1, Cone) +``` + ### Plotting F(x,y, z) = c diff --git a/quarto/differentiable_vector_calculus/vector_valued_functions.qmd b/quarto/differentiable_vector_calculus/vector_valued_functions.qmd index 06302c5..2d9b37c 100644 --- a/quarto/differentiable_vector_calculus/vector_valued_functions.qmd +++ b/quarto/differentiable_vector_calculus/vector_valued_functions.qmd @@ -232,6 +232,7 @@ plot_parametric(0..2pi, f, legend=false) scatter!([0],[0], markersize=4) ``` + ##### Example diff --git a/quarto/index.qmd b/quarto/index.qmd index 41777fe..f8500ae 100644 --- a/quarto/index.qmd +++ b/quarto/index.qmd @@ -11,7 +11,7 @@ This is a set of notes for learning programming language with an easy to learn syntax that is well suited for this task. -Read "[Getting started with Julia](./misc/getting_started_with_julia.html)" to learn how to install and customize `Julia` for following along with these notes. Read "[Julia interfaces](.misc/julia_interfaces.html) to review different ways to interact with a `Julia` installation. +Read "[Getting started with Julia](./misc/getting_started_with_julia.html)" to learn how to install and customize `Julia` for following along with these notes. Read "[Julia interfaces](./misc/julia_interfaces.html) to review different ways to interact with a `Julia` installation. Since the mid 90s there has been a push to teach calculus using many different points of view. The diff --git a/quarto/integral_vector_calculus/double_triple_integrals.qmd b/quarto/integral_vector_calculus/double_triple_integrals.qmd index 07cf8d6..f44bd4f 100644 --- a/quarto/integral_vector_calculus/double_triple_integrals.qmd +++ b/quarto/integral_vector_calculus/double_triple_integrals.qmd @@ -716,6 +716,9 @@ endpoints(ys, x) = callf.(ys, Ref(x)) xs) ``` +The one issue with the above is the tolerances should increase for each outer integration. Otherwise, a certain precision is expected for a noisier integrand. This isn't an issue for many integrands, but can become one. + + ##### Example diff --git a/quarto/integrals/area.qmd b/quarto/integrals/area.qmd index bb33919..80558f7 100644 --- a/quarto/integrals/area.qmd +++ b/quarto/integrals/area.qmd @@ -608,26 +608,23 @@ This value is about $10^{-5}$ off from the actual answer of $1/3$. We should expect that larger values of $n$ will produce better approximate values, as long as numeric issues don't get involved. -Before continuing, we define a function to compute the Riemann sum for us with an extra argument to specifying one of four methods for computing $c_i$: +Before continuing, we define a function to compute Riemann sums for us with an extra argument to specifying one of four methods for computing $c_i$: ```{julia} #| eval: false -function riemann(f::Function, a::Real, b::Real, n::Int; method="right") - if method == "right" - meth = f -> (lr -> begin l,r = lr; f(r) * (r-l) end) - elseif method == "left" - meth = f -> (lr -> begin l,r = lr; f(l) * (r-l) end) - elseif method == "trapezoid" - meth = f -> (lr -> begin l,r = lr; (1/2) * (f(l) + f(r)) * (r-l) end) - elseif method == "simpsons" - meth = f -> (lr -> begin l,r=lr; (1/6) * (f(l) + 4*(f((l+r)/2)) + f(r)) * (r-l) end) - end - - xs = range(a, b, n+1) - pairs = zip(xs[begin:end-1], xs[begin+1:end]) # (x₀,x₁), …, (xₙ₋₁,xₙ) - sum(meth(f), pairs) - +riemann(f, a, b, n; method="right") = riemann(f, range(a,b,n+1); method=method) +function riemann(f, xs; method="right") + Ms = (left = (f,a,b) -> f(a), + right= (f,a,b) -> f(b), + trapezoid = (f,a,b) -> (f(a) + f(b))/2, + simpsons = (f,a,b) -> (c = a/2 + b/2; (1/6) * (f(a) + 4*f(c) + f(b))), + ) + _riemann(Ms[Symbol(method)], f, xs) +end +function _riemann(M, f, xs) + xs′ = zip(Iterators.drop(xs, 1), Iterators.rest(xs, 1)) + sum(M(f, a, b) * (b-a) for (a,b) ∈ xs′) end ``` diff --git a/quarto/integrals/area_between_curves.qmd b/quarto/integrals/area_between_curves.qmd index e78c17c..fd9e05f 100644 --- a/quarto/integrals/area_between_curves.qmd +++ b/quarto/integrals/area_between_curves.qmd @@ -276,6 +276,20 @@ find_zero(m -> area(m) - 1, (0, 1)) (Which is a nice combination of using `find_zeros`, `quadgk` and `find_zero` to answer a problem.) +##### Example + +In an early 2023 article appearing in the [New York Times](https://www.nytimes.com/2023/02/02/opinion/covid-pandemic-deaths.html) a discussion on *excess* deaths was presented. @fig-excess-deaths shows two curves from which the number of excess deaths can be computed. + +::: {#fig-excess-deaths} +![Excess deaths](./figures/excess-deaths.png) + +Illustration of excess deaths. Figure from a Feb. 2023 New York Times article +::: + + +Consider the curve marked *Actual deaths*. The number of deaths per year is the sum over each day of the number of deaths per each day. Approximating this number with a curve and setting 1 day equal to 1 unit, the number of deaths is basically $\int_0^{365} d(t) dt$. This curve is *usually*, say, $u(t)$, so the expected number of deaths would be $\int_0^{365} u(t) dt$. The difference, $\int_0^{365} (d(t) - u(t))dt$ is interpreted as the number of *excess deaths*. This methodology has been used to estimate the true number of deaths attributable to the COVID-19 pandemic. + + ##### Example @@ -740,11 +754,11 @@ Doubling the answer above gives a value that Galileo had struggled with for many #| echo: false imgfile="figures/companion-curve-bisects-rectangle.png" caption = """ -Roberval, avoiding a trignometric integral, instead used symmetry to show that the area under the companion curve was half the area of the rectangle, which in this figure is ``2\\pi``. +Roberval, avoiding a trignometric integral, instead used symmetry to show that the area under the companion curve was half the area of the rectangle, which in this figure is `2pi`. 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Find the length of the hypotenus ##### Example -An article in the [Washington Post](https://www.washingtonpost.com/climate-environment/2022/09/19/ants-population-20-quadrillion/) describes estimates for the number of ants on earth. +A overview of a research paper published in [theconversation.com](https://theconversation.com/earth-harbours-20-000-000-000-000-000-ants-and-they-weigh-more-than-wild-birds-and-mammals-combined-190831) reviews six authors' work on estimating the number of ants currently on earth. This was covered in an +article in the [Washington Post](https://www.washingtonpost.com/climate-environment/2022/09/19/ants-population-20-quadrillion/). -They describe the number of ants two ways: +The authors describe the number of ants two ways: -* There are $20$ *trillion* ants. -* There are $12$ megatons of ants +* There are $20 \cdot 10^{15}$ (20 *quadrillion* or 20 thousand million millions) ants. -With this how many ants make up a pound? +* There is an estimated total biomass of $12$ megatons of *dry* carbon. -Below we use the underscore as a separator, which is parsed as commas are to separate groups of numbers. The calculation is the number of ants divided by the number of pounds of ants (one megaton is $1$ million pounds): + +The authors note in a supplement to their paper that over 15,700 species and subspecies of ants have been named. To get a good estimate, 489 studies of ant populations were combined spanning all continents and major habitats. The studies identify the number of *foraging* ants and combined yield the estimates for the *epigaeic ant abundance* ($3.02 \cdot 10^{15}$) and the *arboreal ant abundance* ($1.34\cdot 10^{15}$). Using an estimate of 22% of ants in a colony are foraging, they get the following estimate for the number of ants: ```{julia} -20_000_000_000_000_000 / (1_000_000 * 12 * 2000) +(3.02 * 10^15 + 1.34*10^15) * 100 / 22 ``` -Or not quite a million per pound. - -Using a pound is $2.2$ killograms or $2,200$ grams, we can this many ants per gram: +Shifting the decimal point, this gives a value rounded to $20\cdot 10^{15}$ ants. +The authors used a value for the *dry weight* of an average (and representative) single ant. What was that value? (Which they indicate is perhaps unreliable, +as, for example, small-bodied ants may be much more abundant than large-bodied ants). We assume below that one "megaton" is $1$ million *metric* tons; a metric ton is $1,000$ kilograms; and a kilogram $1,000$ grams: ```{julia} -20_000_000_000_000_000 / (1_000_000 * 12 * 2000) / 2200 +(12 * 1_000_000 * 1_000 * 1_000) / 20_000_000_000_000_000 ``` -Such combinations will be easier to check for correctness when variable names are assigned the respective values. +Which translates to an *average* dry *carbon* weight of $0.6/1000$ grams, that is $0.6$ milligrams ($0.62$ mg C was actually used). + + The authors write that insects are generally considered to have a dry weight of 30% wet weight, and a carbon weight of 50% dry weight, so the weight in grams of an *average* living ant would be multiplied by $2$ and then $10/3$: + +```{julia} +(12 * 1_000_000 * 1_000 * 1_000) / 20_000_000_000_000_000 * (2 * 10/3) +``` + +That is 4 milligrams, or 250 ants per gram on average. + +Numeric combinations, as above, will be easier to check for correctness when variable names are assigned to the respective values. + +Using the underscore, as above, to separate groups of digits, is helpful, as an alternate to scientific notation, when working with large numbers. + @@ -1094,7 +1108,21 @@ answ=1 radioq(choices, answ) ``` +##### Question +[Sagan's](https://en.wiktionary.org/wiki/Sagan%27s_number) number is defined to be the total number of stars in the observable universe. How big is it? A *sextillion* is 7 groups of three 0's after a leading 1. One estimate is $10$ sextillion. How might this be entered into `Julia`? Select the one that *doesn't* work: + +```{julia} +choices = [ +"`10*10^21`", +"`10*10.0^21`", +"`10e21`", +"`1e22`", + "`10_000_000_000_000_000_000_000`"] +explanation = "With an *integer* base, `10^21` overflows. For typical integers, onmly `10^18` is defined as expected. Once `10^19` is entered the mathematical value is larger the the `typemax` for `Int64` and so the value *wraps* around. The number written out with underscores to separate groups of 0s is parsed as an integer with 128 bits, not 64." +buttonq(choices, 1; explanation=explanation) + +The estimate of 10 sextillion for Sagan's number was made in 1980, a more modern estimate is 30 times larger. ##### Question diff --git a/quarto/precalc/figures/swale.png b/quarto/precalc/figures/swale.png new file mode 100644 index 0000000000000000000000000000000000000000..9a23eac063e1688a867bf151590e60251ff5eab5 GIT binary patch literal 10385 zcmbW7byQT*`|bgyL1JhS>68+XZW!tAZUjV11cr1F9B@EDTBJcrkdho=Bm`+BmF|!( zC4ZamZ><~a{&VkTt(i4(&OV1Tdq3~W^42-IGgqL>M;61*Vs+lha1~u931M>&RksJmF z7l)d%f?DxM|*kxp_K30pmjm0C8L{}#3Db?)Lsp+q(}T2FH5!V^ECx%Nk` zRVYN@Y}PIO!42GaA3DTUqN^*WWMHE-1jTNk+My`cjhL@;9l{J$_bv*1)@> zz$o=nv$C)?9yjpvuQ$rM$V(JkVpUQ#)40U({CE?_SDW)Bi3eBGdr28ZW!;mHeuYz3 z_t>d1cUNg^&$7lj*8Cg3g=105#uvMeEwMO zZ&CmJ^Q3+1y3OU^<#*e4^NoHv+6+vcmy;al@_Vfp&rwuui0jdJvgfLm*y>}iCQ1yR zRIU=AO=+a^k6WNsj!h=YOmr$&qwWwJ6Ebe=*Enlk2}p5Pt!FYbNc3=K1t?Z_Rl1Fr z!UiO|lO5F`ID#vDoN-|()gu#hx*>XY@NJp0Ra{lhRsnXGaFkWrDE^c7$5Hodh=D?u zbfA8zNo7Sn7u{2w+Lt@O5X~@Uo+5aywMB(ZoAp(}2K#5G*$UT_wd)4i{(vvV+pawxDV-Gk;z5G*LZx8Wz{UXmA!T8{_dEKZ* z>m9Ad{mQ&q4&2`Hsw!+L?PexbbkWW~oXLa9^&)(<^?)#B59V>X{_ z7$hTnL>~S0Y)!<0fLu%qoad;;TKj-So!j_vN~1J&!p8XClPrE!L)E|7z6DRdFv-BN zH!&Y7lh`uDFbGHPm?9Y46^}7=D3?-gLQAZlIy_tt>GWGuZ{%1Hi`Z9bnwi2n%L z4Ix66s-PwlMijGUqo5E+C#vqLC!HA_7kh5=ZTKyvyB~q1v`}hG>ODvb4h2(`cXt3r z0?{Uey5=XPw^V#A8P0Eg2&rCeoaq&+Q=%wQX%0lfqC<}kj7*PT-57q}lD z&5~q3l)tn{E8wpzc|fj@2*)-3-F=YWufLi_rfAa@KNUR zz5McsZGkUP%k$mly)hj;QJDm4!qL}QuN!U%i1x_1TMVC%6(`K8dlXSHiJzzTe-P8D zjEp*!42;!wl6+4<%5de*a$lc3Hu$K?<8iVpHeTGd%1FPzkjF2PCpb#}1}p^o!?{W@ zzL#n;pNHd5|GHK(@W@BH8fs&Wx!Q~p&(Cn0D z$jLZtKu+2OQK+7r*mkw!Svrzp6Y!W^t9)6LV(311l)|gae@A86p$ok~MH8dzS~}$> z8A!m9{(D9%N7)LJ)56iJj(9k%3CuLe+YjhU6(b3zJ&;0a-F1`JWfOd>KazjCuTT;} zl~5PE&C^x^^de80YFvf}8K2>xWU2Yg3nqG{Eb*bWe3z@KHpf&dTQR&9@kV9Fy`T(L zeywcU;+bv5{MB3LP(7tX3|UUg0vvPuhf8c{&Xkz?oaLR@DVVfol6qTKTITEOm|Jant-j z5>nSwX$pGxvZwZB-(0#(H+bjE@(wJ+oTWi|8~bcwlJd|j?`t&q>UL$Du~Ha=xR<=g z1&eT&xVIvQc4l$Ck42+TVbdy$deLmNe_xT-hhyXC0xkP+1_>W!&&Hvu{g>w*z zyAIWCd^CJdFOdRqz6<&-SpiGR#GzZu7L-d+XY_H6UwBh{i+E zKw6kxCC_udH4ZyB5>4}(GO0f)={@D^=(LC|G!3-=`Ro|=-lvgSZgI_bZ+tPW^2ymJ zY=i%;rY>i_8~tlrd;DiwcoAE{d20^U%o8-j$8IiyMl9;i1s`cmH~UxiXNdCdnuQv0 z(C`JwMZ;Zx7RAmfS&ujRx)@D7pLn25hfM3Ayyma=-<#8Ke(8at{E$Ql)d0T6NTKVH zKe|84MInhVi+GmGRS^8_`j4MY<u7b`>(FGn2?Cm$+l}F z&qy(!6;&LknD%H-byD+SdgMl)ewDq7Sz9P7Bj<72&hPp+M5A8N*qHJ{*+vr9J9*Et zf)CYD-x*qoyIX9Veg-^P8sWpiey8`1tI><|b#=fNw{P=^5RFoMgQRx4pdDP2Y-_9} z|1933{2;-r-d+;NfwUp=>iDP5Ns4cgW_q4BkwLf2K{f1Tt7MO^!?J(?Y!HDj!yWK1cJCz+QP{hG<^HdY1v&C7&{MXZEBw#pKPYC3g9SSt#AR$u z3f8LGo*JL4tLtb>pl{&mw95&jKRX}I!64~p7dRg!>)skP$B2XuBbN;(`qTTegS3nK)p#H z!6$;!_mkMuX^Onv?9loi_3ywzknVX`45iP>xJlWGiFKgL^;z3ZiQge3PW~b(G7anZ zy}gs3Ak)$)B|&T70Z|w)A^hN5Bq8i=3mpAaToXN4=Wf!t^YypkXLhWSmB3Bzmqbt& zB|(~KBD5{rqq`9aJ*5AyAB@P;M2yKgK#H{Mhb&(0sB*q597EfR3X~37NJE?~7Zxhvzne*?+32(SV_M0B3GXc>Z;_me^*39Q zF<@E*fWD{r&xe@Iezo6-W!FgQDXVPdxjH}jnav__E8Lw-DlR*}>U{e6>Wc&KB-|Go zz<}=an^a`}nL}BG#pKzY2X3ef8kgv+RO*R%xx)^+sJB0tTTGI(t8ehdP~n_BO5a8! 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Doing so makes it much easier to check for accuracy against the expression to compute. +##### Example + +A [grass swale](https://stormwater.pca.state.mn.us/index.php?title=Design_criteria_for_dry_swale_(grass_swale)) is a design to manage surface water flow resulting from a storm. Swales detain, filter, and infiltrate runoff limiting erosion in the process. + +![Swale cross section](precalc/swale.png) + +There are a few mathematical formula that describe the characteristics of swale: + +The area is given by: + +$$ +A = (b + d/\tan(\theta)) d +$$ + +The *wetted* perimeter is given by + +$$ +P = b + 2 d/\sin(\theta) +$$ + +The *hydraulic radius* is given by + +$$ +R = \frac{b\cdot d \sin(\theta) + d^2 \cos(\theta)}{b\sin(\theta) + 2d}. +$$ + +Finally, the *flow quantity* is given by *Manning's* formula: + +$$ +Q = vA = \frac{R^{2/3} S^{1/2}}{n}, \quad R = \frac{A}{P}. +$$ + +With $n$ being Manning's coefficient, $v$ the velocity in feet per second, and $S$ being the slope. Velocity and slope are correlated. + +Manning's coefficient depends on the height of the vegetation in the grass swale. It is $0.025$ when the depth of flow is similar to the vegetation height. + +Given all this, compute the flow quantity when $S = 2/90$, $n=0.025$ and $v=1/10$ for a swale with characteristics $b=1$, $\theta=\pi/4$, $d=1$. + +```{julia} +b, theta, d = 1, pi/4, 1 +n, S = 0.025, 2/90 +A = (b + d/tan(theta)) * d +P = b + 2d/sin(theta) +R = A / P +Q = R^(2/3) * S^(1/2) / n +``` + + ##### Example