diff --git a/lamv.typ b/lamv.typ index baace03..4987b0f 100644 --- a/lamv.typ +++ b/lamv.typ @@ -1,10 +1,13 @@ #import "@local/evan:1.0.0":* #let poonen = link("https://math.mit.edu/~poonen/notes02.pdf")[Poonen's notes] -#let epigraph(quote_src, quote_text) = pad(left: 35%)[ - #emph(quote_text) \ - #line(length: 100%) - #align(right)[--- #quote_src] -] +#let epigraph(quote_text, attribution: none) = { + set text(size: 10pt) + pad(left: 35%)[ + #emph(quote_text) + #line(length: 100%) + #if attribution != none { align(right)[--- #attribution] } + ] +} #show: evan.with( title: [Linear Algebra and Multivariable Calculus], @@ -28,6 +31,12 @@ #pagebreak() #part[Part Alfa: Linear Algebra of Vectors] +#epigraph(attribution: [_Infinite Jest_, by David Foster Wallace])[ + It's a very American illness, + the idea of giving yourself away entirely to the idea + of working in order to achieve some sort of brass ring + that usually involves people feeling some way about you. +] For comparison, Part Alfa corresponds roughly to §1, §2, §3.9 of #poonen. #include "src/vectors.typ" #pagebreak() @@ -39,6 +48,14 @@ For comparison, Part Alfa corresponds roughly to §1, §2, §3.9 of #poonen. #pagebreak() #part[Part Bravo: Linear Algebra of Matrices] +#epigraph(attribution: [Lydia's opening line in + #link("https://anserena.com/other/writing/")[_Holding on to Her Words_], by Serena An])[ + Translation is a bridge. It can bridge two languages, time periods, cultures. + The pages of a book and your mind. Even the space between two people. + Translation involves reflecting on yourself and your relationships with others. + That’s at the heart of why I study Chinese literature. + But it took me many years to realize that. +] For comparison, Part Bravo corresponds roughly to §3, §4, §6 of #poonen. #include "src/matrix.typ" #pagebreak() @@ -48,6 +65,9 @@ For comparison, Part Bravo corresponds roughly to §3, §4, §6 of #poonen. #pagebreak() #part[Part Charlie: Review of complex numbers] +#epigraph(attribution: [#link("https://w.wiki/Tpof")[_Stephen's Sausage Roll_]])[ + Learned people cast great shadows. +] For comparison, Part Charlie corresponds roughly to §11 of #poonen. #include "src/complex.typ" #pagebreak() @@ -55,11 +75,21 @@ For comparison, Part Charlie corresponds roughly to §11 of #poonen. #pagebreak() #part[Part Delta: Parametric side-quest] +#epigraph(attribution: [Koro-sensei in _Assassination Classroom_])[ + Broadly speaking, there are two reasons a person feels the desire to teach something: + Either he wants to pass on his successes or he wants to pass on his failures. +] For comparison, Part Delta corresponds roughly to §5 and §7 of #poonen. #include "src/param.typ" #pagebreak() #part[Part Echo: Multivariable differentiation] +#epigraph(attribution: [#link("https://www.ams.org/notices/201005/rtx100500608p.pdf")[What is mathematics for?], by Underwood Dudley])[ + However, when I am before a bar of judgment, heavenly or otherwise, + and asked to justify my life, I will draw myself up proudly and say, + "I was one of the stewards of mathematics, and it came to no harm in my care". + I will not say, "I helped people get jobs". +] For comparison, Part Echo corresponds roughly to §8 and §12.1-§12.3 of #poonen. #include "src/level.typ" #pagebreak() @@ -71,6 +101,9 @@ For comparison, Part Echo corresponds roughly to §8 and §12.1-§12.3 of #poone #pagebreak() #part[Part Foxtrot: Optimization] +#epigraph(attribution: [Queen in #link("https://w.wiki/6WCE")[_Deltarune_], by Toby Fox])[ + Teens Are Merely Big Children #text(size:7pt)[And Adults Are Even Bigger Children] +] For comparison, Part Foxtrot corresponds roughly to §9 and §12.4-§12.6 of #poonen. #include "src/minmax.typ" #pagebreak() @@ -82,6 +115,10 @@ For comparison, Part Foxtrot corresponds roughly to §9 and §12.4-§12.6 of #po #pagebreak() #part[Part Golf: 2D integrals of scalar functions] +#epigraph(attribution: [Calvin in _Calvin and Hobbes_ (May 3, 1989)])[ + Mom says Hobbes wouldn't have stolen because he's not valuable. \ + ... (sniff) well, *I* think he's valuable. +] For comparison, Part Golf corresponds to §13.1-§13.5 of #poonen. #include "src/ipep.typ" #pagebreak() @@ -95,6 +132,15 @@ For comparison, Part Golf corresponds to §13.1-§13.5 of #poonen. #pagebreak() #part[Part Hotel: 3D integrals of scalar functions] +#epigraph(attribution: [Author notes for + #link("https://puzzles.mit.edu/2023/puzzlefactory.place/puzzles/5d-barred-diagramless-with-multiverse-time-travel")[5D Barred Diagramless with Multiverse Time Travel], + by Alex Irpan])[ + At some points, I was not convinced this puzzle should even exist, + but editors and testsolvers encouraged me to finish it, saying it was + "exactly the bullshit I expect to see in Mystery Hunt". + (The lucky postprodder for the puzzle would also like to voice his support.) + Hope you had fun! Really, it could have been much worse. +] For comparison, Part Hotel corresponds to §13.6-13.9 and §17.1-§17.6 of #poonen. #include "src/triple.typ" #pagebreak() @@ -108,6 +154,12 @@ For comparison, Part Hotel corresponds to §13.6-13.9 and §17.1-§17.6 of #poon #pagebreak() #part[Part India: Line integrals of vector fields over a curve] +#epigraph(attribution: [EWD 1036, by Edsger W. Dijkstra (1988)])[ + My point today is that, if we wish to count lines of code, + we should not regard them as "lines produced" but as "lines spent": + the current conventional wisdom is so foolish as to book + that count on the wrong side of the ledger. +] For comparison, Part India corresponds to §14, §15, §20, §21 of #poonen. #include "src/vecfield.typ" #pagebreak() @@ -125,6 +177,10 @@ For comparison, Part India corresponds to §14, §15, §20, §21 of #poonen. #pagebreak() #part[Part Juliett: Flux integrals of vector fields over a surface] +#epigraph(attribution: [Monokuma in _Danganronpa_])[ + Every human has regrets, has things they'd like to go back and change. + But I don't! 'Cause I'm a bear! +] For comparison, Part Juliett corresponds to §17.7, §18, §19, §21 of #poonen. #include "src/flux.typ" #pagebreak() @@ -134,10 +190,17 @@ For comparison, Part Juliett corresponds to §17.7, §18, §19, §21 of #poonen. #pagebreak() #part[Part Kilo: Practice questions] +#epigraph(attribution: [Lester Gardner])[ + The beaver is noted for his engineering, mechanical skills, and industry. + His habits are nocturnal. He does his best work in the dark. +] #include "src/practice.typ" #pagebreak() #part[Part Lima: Solutions] +#epigraph(attribution: [#link("https://mitadmissions.org/discover/life-culture/hacks-tomfoolery-pranks/")[MIT hacker code of ethics]])[ + Brute force is the last resort of the incompetent. +] #include "src/sol-alfa.typ" #pagebreak() #include "src/sol-bravo.typ" @@ -162,4 +225,10 @@ For comparison, Part Juliett corresponds to §17.7, §18, §19, §21 of #poonen. #pagebreak() #part[Part Mike: Appendix] +#epigraph[ + You all have a little bit of "I want to save the world" in you, + that's why you're here, in college. + I want you to know that it's okay if you only save one person, + and it's okay if that person is you. +] #include "src/appendix.typ" diff --git a/recitations/r03.typ b/recitations/r03.typ index c39bff8..05c9e87 100644 --- a/recitations/r03.typ +++ b/recitations/r03.typ @@ -7,7 +7,7 @@ date: [11 September 2024], ) -#quote(attribution: [Monokuma in _Danganropa_])[ +#quote(attribution: [Monokuma in _Danganronpa_])[ Every human has regrets, has things they'd like to go back and change. But I don't! 'Cause I'm a bear! ] diff --git a/src/divthm.typ b/src/divthm.typ index 9ed1eca..aa9831e 100644 --- a/src/divthm.typ +++ b/src/divthm.typ @@ -185,9 +185,9 @@ We can jump straight into examples now! Obviously $int_(theta=0)^(2pi) dif theta = 2pi$. The double integral can be evaluated as $ int_(r=0)^1 r int_(z=0)^(r^2) z dif z dif r - &= int_(r=0)^1 r dot [z^2/2]_(z=0)^(r^2) dif r \ - &= int_(r=0)^1 r^5/2 dif r \ - &= [r^6/12]_(r=0)^1 = 1/12. $ + = int_(r=0)^1 r dot [z^2/2]_(z=0)^(r^2) dif r + = int_(r=0)^1 r^5/2 dif r + = [r^6/12]_(r=0)^1 = 1/12. $ Hence the final answer is $ 1/12 dot 2 pi = #boxed[$ pi / 6 $]. #qedhere $ ] diff --git a/src/flux.typ b/src/flux.typ index edc5464..d8fc5db 100644 --- a/src/flux.typ +++ b/src/flux.typ @@ -75,10 +75,6 @@ in addition to the actual points, we need to specify an _orientation_. To be more precise, at every point $P$ of the surface $cal(S)$, the cross product from our parametrization could point in one of two opposite directions. -#definition(title: [Definition of orienting a surface])[ - To _orient_ the surface $cal(S)$ is to specify, at each point, - which way you want the cross product of your parametrization to point. -] #figure( image("figures/flux-cross-sign.svg", width: auto), @@ -89,6 +85,11 @@ the cross product from our parametrization could point in one of two opposite di ], ) +#definition(title: [Definition of orienting a surface])[ + To _orient_ the surface $cal(S)$ is to specify, at each point, + which way you want the cross product of your parametrization to point. +] + Algebraically, this corresponds to choosing the _order of $u$ and $v$_; as if you flip the order of the two parameters it will negate the entire cross product: $ (partial bf(r))/(partial u) times (partial bf(r))/(partial v) diff --git a/src/ipep.typ b/src/ipep.typ index fcc5f62..769653c 100644 --- a/src/ipep.typ +++ b/src/ipep.typ @@ -11,6 +11,14 @@ If you've been following my advice to pay attention to type safety so far, it'll I'll freely admit that I (Evan) often make type-errors in this part of 18.02 as well, so don't let your guard down. +#figure( + image("figures/integrals-triangle.svg", width: 100%), + caption: [ + For each $0 <= d <= n <= 3$, we draw the kind of integral and give it a name. + Download at #url("https://web.evanchen.cc/textbooks/poster-ints.pdf"). + ] +) + Remember that: #idea[ Suppose $f : RR^n -> RR$ is given, and $0 <= d <= n$. @@ -25,19 +33,10 @@ So philosophically, a $0$-dimensional integral is just a finite sum of $f$ at so This might seem stupid that I bring up this degenerate case, but it turns out later when we cover div/grad/curl the $0$-dimensional case is relevant. -Here's a giant chart in @poster-triangle. +We give a giant chart in @poster-triangle. (The chart is so big it doesn't quite fit in the page, but you can download a #link("https://web.evanchen.cc/textbooks/poster-ints.pdf")[large PDF version]). - -#figure( - image("figures/integrals-triangle.svg", width: 100%), - caption: [ - For each $0 <= d <= n <= 3$, we draw the kind of integral and give it a name. - Download at #url("https://web.evanchen.cc/textbooks/poster-ints.pdf"). - ] -) - This chart has ten different kinds of integrals, one for each $(d,n)$ with $0 <= d <= n <= 3$. Here's a rundown of the things in the chart. diff --git a/src/practice.typ b/src/practice.typ index 5b10daf..122c8e3 100644 --- a/src/practice.typ +++ b/src/practice.typ @@ -54,8 +54,8 @@ Solutions are in @ch-sol-kilo. with normal vector oriented outwards (thus $cal(S)$ is the curved part of a cylinder). Compute the divergence of $bf(F)$. Then compute the flux of $bf(F)$ through $cal(S)$. - (Recommended approach: add two "lids" to $cal(S)$, calculate flux through the lids by hand, - then use the divergence theorem.) + (Recommended approach: add two "lids" to $cal(S)$ + and use the divergence theorem.) ] == Miscellaneous practice questions without solutions diff --git a/src/sol-alfa.typ b/src/sol-alfa.typ index 0204d9e..03e7e96 100644 --- a/src/sol-alfa.typ +++ b/src/sol-alfa.typ @@ -140,10 +140,10 @@ $ op("comp")_(bf(w))(bf(v)) = frac(bf(v) dot bf(w), |bf(w)|) The vector projection of $bf(v)$ along $bf(w)$ is given by: $ op("proj")_(bf(w))(bf(v)) - &= frac(bf(v) dot bf(w), |bf(w)|) (bf(w) / (|bf(w)|)) \ - &= -11/5 dot vec(-3000, -4000, 0) / 5000 \ - &= -11/5 dot vec(-3 slash 5, -4 slash 5, 0) \ - &= #boxed[$ vec(33 slash 25, 44 slash 25, 0) $]. $ + = frac(bf(v) dot bf(w), |bf(w)|) (bf(w) / (|bf(w)|)) + = -11/5 dot vec(-3000, -4000, 0) / 5000 + = -11/5 dot vec(-3 slash 5, -4 slash 5, 0) + = #boxed[$ vec(33 slash 25, 44 slash 25, 0) $]. $ == Solution to @exer-dot-reverse (dot product $3$) @@ -156,6 +156,13 @@ See @fig-sol-alfa-dot-reverse for a picture. (You might already guess one of the solutions --- $bf(v) = chevron.l 1,0 chevron.r $ obviously works --- but we'll pretend we didn't notice that.) +#figure( + image("figures/sol-alfa-dot-reverse.svg", width: auto), + caption: [The two answers to @exer-dot-reverse, + which are two unit vectors spaced at an equal angle away from $chevron.l 3,4 chevron.r$. + ], +) + Translating the givens algebraically, we have the following system of equations: $ bf(v) dot bf(w) = 3 &==> 3 x + 4 y = 3 \ @@ -180,13 +187,6 @@ In conclusion the answer is $ #boxed[$ chevron.l 1,0 chevron.r " and " lr(chevron.l -7/25, 24/25 chevron.r) $]. $ See @fig-sol-alfa-dot-reverse for a picture of the two answers. -#figure( - image("figures/sol-alfa-dot-reverse.svg", width: auto), - caption: [The two answers to @exer-dot-reverse, - which are two unit vectors spaced at an equal angle away from $chevron.l 3,4 chevron.r$. - ], -) - == Solution to @exer-cauchy ($2$ and $5$) #recall-thm() diff --git a/src/triple.typ b/src/triple.typ index e181da4..6afaac9 100644 --- a/src/triple.typ +++ b/src/triple.typ @@ -82,19 +82,16 @@ so one needs different letters. 2sqrt(1-x^2) int_(y=-sqrt(1-x^2))^(sqrt(1-x^2)) 1 dif y dif x \ &= int_(x=-1)^1 - 2sqrt(1-x^2) dot 2sqrt(1-x^2) dif x \ - &= int_(x=-1)^1 4(1-x^2) dif x \ + 2sqrt(1-x^2) dot 2sqrt(1-x^2) dif x + = int_(x=-1)^1 4(1-x^2) dif x \ &= 4[x - x^3 / 3]_(x=-1)^1 = 4[(1 - 1 / 3) - (-1 + 1/3)] = #boxed[$ 16 /3 $]. #qedhere $ ] #digression(title: [Digression on picture])[ If you draw a picture of the region, you get the intersection of these two cylinders which forms something apparently called a _Steinmetz solid_. - (I say "apparently" because ChatGPT told me this name; - I didn't know this had a name before either.) - Surprisingly, you actually _don't_ want to use polar (or cylindrical) coordinates - on this example. + Surprisingly, you actually _don't_ want to use polar (or cylindrical) coordinates on this example. If you try to do so, I think you'll actually get stuck. Straight $x y z$-integration turns out to work because of the unexpectedly convenient fact that you get two square roots that miraculously cancel. @@ -167,13 +164,12 @@ If you were able to do the earlier example with $z = 3(x^2+y^2)$ and $z = 72-5(x^2+y^2)$ by yourself without reading the solution, then you can safely skip this entire section! -The transition map $ (r, theta, z) |-> (x, y, z)$ is given by +The transition map $(r, theta, z) |-> (x, y, z)$, illustrated in @fig-triple-cylinder, is given by $ x &= r cos theta \ y &= r sin theta \ z &= z. $ -This is illustrated in @fig-triple-cylinder. #figure( image("figures/triple-cylinder.svg", width: auto), caption: [Picture of cylindrical coordinates.