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/sci/ - Science & Math

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10215302 No.10215302 [Reply] [Original]

What do your various undergrad calc/analysis courses contain?
I see people say that rigor is often spared until later years, so what is left out?
So far we've had to study proofs of most theorems with a few exceptions like the intermediate and extreme value theorems.
In middle of junior year doing engineering physics, here's my calc so far:

>"Introductory mathematical analysis"
>High school reminders like trig&polynomials
>Semi-serious construction of integers, rationals and reals (proof of basic properties was considered too tedious)
>Induction
>Limits and continuity with ε-δ
>Differentiation and antidifferentiation
>The Riemann integral
>Relevant theorems for proof of FTC
>Generalized integrals

>"Real analysis"
>Linear and Euler differential equations
>Taylor polynomials and representations of the rest term
>L'Hospital
>Difference equations
>Iteration, Newton-Raphson
>Banach fixed-point theorem
>Series and convergence tests (integral, direct and limit comparison, ratio and root)
>Absolute convergence, alternating series and Leibniz criterion
>Power series, Taylor series and radius of convergence
>Function sequences and series
>Pointwise vs. uniform convergence
>Uniform limit, limits under integral sign, M-test (Currently here in the course)
>Dominated convergence, changing order of limits
>Sets in R^n
>Limits and continuity for functions of multiple variables

Further courses are multivariable, complex and Fourier analysis.

>> No.10215320

>>10215302
>antidifferentiation

indefinite integrals you fucking retard

>> No.10215325

>>10215302
also we are not your counselors fuk off

>> No.10215477

>>10215320
and an indefinite integral is when you....antidifferentiate
fucking mong

>> No.10215483

>>10215302
Is this loss?

>> No.10215493

>>10215477
>fucking mong
Says the retard that would rather use primal terms. Gtfo w your non rigorous vocabulary neanderthal

>> No.10215505

>>10215493
>"y-yeah...well...you used a primal term!"
thanks for the laugh

>> No.10215552

>>10215302
I skipped analysis so I have no fucking clue dude. Graduating tomorrow and I haven't done a single epsilon-delta proof in my life.

>> No.10215603

>>10215505
how about I antimultiply your mom into two pieces with my cock

>> No.10215685

>>10215505
> "an indefinite integral is when y-you.. anti differentiate!"
Fell right for my bait ezpz

>> No.10215697

>>10215320
"Indefinite integrals" have little to do with integration and everything with antidifferentiation.

>> No.10215731

Analysis (for math majors, all proof based) in German unis

>Analysis 1 (mandatory in the first semester of math b.sc.):
Induction, definition of the reals, basics of complex numbers, sequences, convergence, Bolzano-Weierstraß-Theorem / Cauchy's criterion / completeness, series and all that, partial fraction decomposition, continuous functions R->R, differentiable functions, Taylor series, Riemann integration, Fundamental Theorem of Calculus, sequences of functions / uniform convergence
>Analysis 2 (2nd semester)
metric spaces, limits and continuity in topological spaces, Compact subsets / Heine-Borel theorem, rectifiable curves R->R^n, functions R^n->R^m, partial derivatives, total derivative, Schwarz's theorem, multidimensional Taylor series, extremal values of functions R^n->R, Banach fixed point theorem, implicit function theorem, theorem of the inverse function, Lagrange multipliers, differentiating under the integral sign. Ordinary differential equations: elementary solving strategies, reducing higher order ODEs to systems of 1st order ODEs, Picard-Lindelöf-theorem / -iteration, linear systems of ODEs.
>Analysis 3 (mandatory for math b.sc.)
measure theory, lebesgue integration in R^n and sheiit (I haven't taken it yet)
>Funktionentheorie / Complex Analysis (not mandatory)
holomorphic functions, Riemann-Stieltjes integration, Cauchy's formula and Cauchy's theorem including the homotopic version, liouville's theorem and all thag, fundamental theorem of algebra, morera's theorem, residues, laurent series, schwarz's lemma, Riemann mapping theorem, gamma function

>> No.10215754

>>10215302
In the US we basically never make engineers learn calculus rigorously at all, never do an epsilon-delta proof (or do one trivial one and then never talk about it again).

But claiming that students really study the proofs doesn't mean much unless they're asking you to write proofs on the test (I mean somewhat original proofs, not literally just reciting a proof you were given to memorize).

>> No.10216138

>>10215731
alright, seems like german analysis 1 is basically the same as what i'm doing, then.
>>10215754
that seems strange, any idea why not? might be a matter of prestige, as all the most prestigious scientific degrees in sweden are five-year (bachelor+master) engineering degrees considered the toughest degrees available across all fields.
but does that mean that even at an MIT, CalTech or such you could graduate with a highschool-level grasp of analysis?

>> No.10216151

>>10215302
>L'Hospital

>> No.10216152

>>10215552
If you've nothing to contribute, spare us the reply

>> No.10216182

>Advanced Calculus 1 (our stupid name for undergrad real analysis 1)
Some set theory in the very beginning of the semester, proving deMorgan's and Russell's paradox
Proving some absolute value stuff, triangle inequality
Archimedean Property
Density of the Rationals
Supremums and infimums
Epsilon-Delta definitions for Convergence of sequence, Cauchy sequences, continuity, and differentiability
Bolzano-Weierstrass, Heine-Borel Theorems
Some Topology: open, closed sets, compactness, connectedness
General Metric Spaces, general notion of distance and triangle inequality in them
MVP, Rolle's Theorem
Integration, Darboux sums

Overall it was super AIDS and I'm so grateful that I don't have to take Advanced Calculus 2, or complex analysis for my secondary math ed major. I prefer Algebra and logic way more over Analysis. I would have taken a second semester of Modern Algebra over Advanced Calculus 1 any day.

>> No.10216183

>>10216182
MVT*, kek

>> No.10216184

>>10215302
>L'Hospital
Sopa de macaco uma delícia

>> No.10216222

>>10215302
You know, I often see people talk about their math courses here and it baffles me how basic/non rigorous they are compared to mine when my university isn't even that good by international standards and I'm not even studying pure math.

>> No.10216559

>>10216138
>that seems strange, any idea why not?
Constructing mathematically correct proofs is difficult.
>, as all the most prestigious scientific degrees in sweden are five-year (bachelor+master) engineering degrees considered the toughest degrees available across all fields.
The swedish talent pool is small enough to allow studious brainlets into most Civilingenjör-programs (excluding teknisk fysik and indek, possibly); were KTH to suddenly require proofs in mandatory basic calc courses, half the student body would fail the course ad infinitum.

>> No.10216561

>>10216182
>i prefer algebra and logic
i wonder why