I've already self-studied calculus, so I wanted to go on and learn it in multiple variables. Generally, Apostol seems to have had good reviews for 'pure, prose learning', but how is it to actually learn from it? And if I do should I read the single-variable calculus section?
calculus in a nutshell:
I will save you some time:
There are 4 principles all calculus of one variable is based on.
I. IVT (Intermediate value theorem) If a function f is continuous on an interval, then the values it takes also form an interval.
Hence if there exist points a,b in the domain interval where f(a) < K < f(b), then f(x) must equal K at some c between a and b.
E.g. the continuous function f(x) = x^3 + x + 1 must equal zero for some x between -1 and 0, since f(-1) = -1, and f(0) = 1.
II. EVT (Extreme value theorem, also called Max Min Value or MMV)
If a function is continuous on a closed bounded interval, then its value also form a closed bounded interval, i.e. particular there is a (finite) smallest and a (finite) largest value.
I.e. if f is continuous on [a,b] then there exist c,d in [a,b] such that for every x in [a,b], f(c) ≤ f(x) ≤ f(d).
E.g. since the volume of the right circular cone of height R+x, inscribed in a sphere of radius R, has volume (π/3)(R^2-x^2)(R+x), is continuous for 0 ≤ x ≤ R, some one of these cones has largest volume.
III. (Rolle’s Theorem) If f is continuous on [a,b] and differentiable on (a,b), and if f(a) = f(b), then f achieves its maximum (or minimum) at a point c between a and b where f’(c) = 0, i.e. at a “critical point” for f.
Cor: If f has two derivatives on [a,b] and f’ takes the same value twice, e.g. if there are two critical points, and if f’’ is zero only a finite number of times, then f has a flex somewhere on (a,b) where f’’=0.
Cor: If f is continuous on [a,b] but has no critical points in the interval (a,b), then f cannot take the same value twice in [a,b], hence cannot change direction, i.e. f is strictly monotone on [a,b].
Cor: If f’’ exists but is never zero on [a,b], then f never changes concavity on [a,b], i.e. f is either concave up or concave down on all of [a,b].
IV. MVT (Mean value theorem) If f is continuous on [a,b] and diffble on (a,b), there is a point c with a<c<b, and f’(c)(b-a) = f(b)-f(a).
Cor: If f’=g’ on [a,b] then f-g is constant on [a,b].
Cor: Since for continuous f, d/dx (integral of f from a to x ) = f(x), then integral of f from a to b = G(b)-G(a), for any G with G’ = f.
Haha, wow, I could actually make sense of all these theorems - awesome.
I have once attempted an analysis text and was overwhelmed with the terminology that was apparently essential to calculus - point-set topology and metric spaces? Etc.
horse poop. i have just given you all the necesities.
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