Basic properties of R
Archimedean Property
- dense of Q in R: always can find a rational number between two real numbers
- step functions
least upper/greatest lower bound property
- one concludes the another
- min and max
Trigonometric Inequality
- use in limit proof
irrational numbers
- proof of irrationality of square roots
- proof of existence of natural number-th root of positive real number
Sequence and Series
Sequence
recursive definition
- techniques
Convergence
limit definition
- negation
- limit conservative theorem/squeeze theorem
- Operational rules of limit
- boundedness and convergence
- Monotonic Convergence Theorem
convergence of subsequence
- Nk >= K
- all subsequences are convergent iff convergent
- Nested interval theorem
Cauchy Sequnce
a sequence is cauchy iff it is convergent
- a->b: construction of subsequence/boundedness of cauchy sequence
- b->a:TI
- cauchy iff limSup = limInf
- cauchy sequence are bounded
applications
- approximation
real numbers are limits of cauchy sequences
- definition of real exponents
- the negation
LimInf/ LimSup
- Necessary Condition of Existence
Construction of an and bn
- monotonicity of an/bn
- convergence of an/bn
Construction of subsequence that converge to A/B
necessary condition: bounded
- B-W theorem: all bounded sequences have a convergent subsequence
technique
- |Ank-a|<1/k
conclusions
- weaker limit conservative theorem(very useful)
Series
convergence tests
- Cauchy's Criterion
- necessary: an goes to 0
comparison test
- absolute convergence indicates convergence
- greater than 0 and bounded by the bigger term\, by MCT\, convergent
root test(limSup)
- a<1\, a>1 are informative. a ->infinity is not
ratio tests(limSup)
- a<1 is informative only
- if liminf >1\, diverge
power series
- radius of convergence
e
series definition
- irrationality
limit definition
- limit conservative theorem
Convergence of summation by parts
- alternating series test
Convergence of catchy's Product
- one abs converge/ one converge
Convergence of rearrangement
converge to original value if abs converge
- gema(n) converge to +infinity
Point-set Topology
set theory
- finite/infinite
countable/uncountable
- Z\, Q are countable
- union of countable set is countable
- bijection to N
definition of section and union
sequence of set
- union and intersection: definition
points
definitions
- neighborhood definition
- limit
- boundary
- exterior
- interior
- adherent
- relations
named sets(metric)
axioms of distant function
同一
- d(x\,y) = 0 <-> x= y
- 对称
- 三角
- open/close sets
- bounded/unbounded set
dense set in Metric space
- Q is dense in R by (AP)
compact sets
definition of open cover
- for mark a of each subset\, a can belong to a uncountable set
compact sets are closed
- by picking r(x) < d(x\,p)/2
compact sets are bounded
- use all neighbors of each element to construct an open cover
closed subset of a compact set is compact
- (A^c U open cover of A ) is an open cover of X
- H-B theorem: for a subset of R^n\, it closed and bounded iff it is compact
- techniques involved: chose Nr(p) for p in A
connected/disconnected set
- polygon theorem
a real interval is connected
- connected subset of R is an interval
- non-empty open disjoints
Continuity
epsilon-delta definition in metric space
- must converge to values in the metric space(range)
convergence criterion of general metric space
- construct sequence that converge
- from general space to R^k\, k>=2
- from R^k to R (partial convergence)
continuity
- definition: limit points involved
open/closed set and continuity
- reverse image of open/closed subset of continuous function
connected set and continuity
- intermediate value theorem
compact set and continuity
- extreme value theorem
uniform continuity
- continuity and uniform continuity
uniform continuity and boundedness
- uniformly conti in (a\,b) -> bounded
compactness and uniform continuity
- continuous & compact domain -> uniformly continuous
discontinuity
side-limit
side limit and monotonicity
existence of side limit
- monotonic: both exist
- monotonic: convert to Sup and Inf
inequality of x\, y
set of dis continuity is at most countable
- (AP)/D injective to S\, S injective to Q
two types of discontinuity
- first kind: two side-limits both exist
- second kind: not both exist
Differentiation
derivative
- operational
geometric
- interpretation: error decreases faster than variable
differentiable and continuous
- tools: linear inequality
local extrema
- def: delta language
interior + differentiable: derivative=0
- tool: limit conservative theorem
MVT
- condition: 1)continuous on [a\,b] 2)differentiable on (a\,b)
- tools: 1) Extreme value theorem 2) Construction of single variable function
conclusions
f(a)-f(b)=(a-b)f'(x)
monotonicity iff derivative has fixed sign
- condition: open interval\, differentiable
- constant iff derivative =0
R to R^k type : inequality
- tools: 1) EVT 2) Construction 3) Cauchy's inequality
injection and derivative
- f'(x) = 0: f is injective
L'Hospital's Rule
- tools: MVT\, injection\, LCT
condition
- existence of lim(f'/g')= L (can belong to R or infinity)
- f\, g differentiable on (a\,b)
- g'(x) \!= 0
two cases
- lim f(x)=0\, lim g(x)=0
- lim g(x) = infinity
conclusion
- lim (f/g)(x) = L
Taylor's Formula
- tools:
discontinuity of derivative
- special property: another version of Intermediate value
- discontinuous => second type
Rhimann Integral
conditions
- defined on closed interval
inequality
- U(P\, f) >= L(P\, f)
inf U >= sup L
- taking common refinement
- f> g - > integration aslo true
- upper/ lower bound inequality
operational rules
f+g over I
- sup(A+B)<= SupA + SupB/ infA+B < = infA + infB
(f over I1) + (f over I2) = f over (I1 U I2)
- taking refinement of single point and P
f in R([a\, b]) -> |f| in R([a\, b])
- Sup f - Inf f >= Sup |f| - Inf |f|
f in R([a\, b]) -> f^2 in R([a\, b])
- EVT\, |f|<=M\, Sup f - Inf f >= Sup |f| - Inf |f|
f\, g in R([a\, b]) -> f*g in R([a\, b])
- fg = 1/2(f+g)^2 + 1/2(f-g)^2
integrate by parts
condition
- f(u(x))
- u' and f are integrable
- u is strictly increasing
- conclusion: the validity of integration by parts
tool
- distance interpretation
- bridge between two denotations
MVT for integration
tools
IVT
- f(x) should be continuous
construction for g(x) involving upper/lower bound
- EVT
integrability and
- difference
sequence of partition
- f in R([a\, b]) -> exist (Pn) in Pa\,b s.t limit converge to Riemann integration
monotonicity
monotone -> Riemann integrable
- taking uniform partiton
continuity
defined on compact set (closed interval) -> uniform continuity
- taking Unif partition with length smaller than delta
differentiability
construction of integration funtion
- continuity complement
uniform continuity of F(x)
- by Lipschitz function\, taking k = sup f
condition: f conti at x
- connection between domain and f
- other wise F is not necessarily differentiable
tool: MVT for integration
- |c- x| < delta by theorem
FTC
anti derivative.of a continuous function is the integration
- attention: still holds if f have some 1st type discontinuities
Sequence and Series of functions
- point-wise convergence
uniform convergence
- def and negation
unif convergent(necessary) =>
- every sequence
- norm of uniform convergence->0
continuous f at x
- if fn(x) is conti at x for all n
- interchange of limit sign
- cauchy for every x
calculus
integration
- condition: fn(x) riemann integrable for all n
conclusion
interchange of limit and integrate
- integrate term by term if sum function Sn(x) is unif convergent
tool
- construct the norm of uniform convergence to be epsilon(n)
differentiation
condition(fn(x) needn't be known to be unif convergent)
- at least one point-wise convergence
- fn(x) is differentiable for all n
- derivative of fn(x) is uniform convergent
conclusion
- fn - > f uniformly
- f'n - > f' uniformly
- differentiation term by term
tools
- MVT
- goal: to show quotient converges uniformly
step1 show fn converges unif
- f'n converge point wise
- step2 f'n converges unif
valid conditions for unif convergence(sufficient)
Weierstrass M test
- only for series
- every sequence
- cauchy for every x
- norm of uniform convergence
complete matrix space
complete if every cauchy sequence in it is convergent to a point in that set
- contraction theorem for unique fixed point
power series
- power series and uniform convergence
Weirstrass approximation
condition
- f is continuous on [a\,b](compact set)
conclusion
- there exists a sequence of polynomial functions that uniformly converges to f
Ascoli's Theorem
condition
- compact domain
- uniformly bounded
- uniformly equicontinuous(equicontinuous would be enough\, uniformity given by compactness)
conclusion
- there is a uniformly convergent subsequence of (fn(x))
steps
1. existence of a countable subset of Domain on which there is a uniformly convergent subsequence
- W-B thm: existence of convergent subsequence
- diagonal argument
2. point-wise convergence of that subsequence on the WHOLE Domain
- compactness: finite open subcover with radius < delta
- TI
3. uniformity of that subsequence on the whole Domain
- finite set -> existence of max(Si)
differentiation of multi-variable functions
definition of Differential of a function(Jacobean matrix)
- dimension match: f: n -> m \, Df in n*m
- cauchy inequality for matrix product
differentiability
- original definition
a sufficient condition(for R^n to R functions)
- Df is continuous
continuously differentiable
- original : each entry is continuous
matrix norm
- epsilon-delta
Chain rule
condition
- both functions are differentiable at certain point
conclusion
- differentiable at that point
- D(g(f(a))) = Df(a)*Dg(f(a))
MVT(general case)
condition
convex domain
- euclidian space
- balls in R^n are convex
- bounded differential
conclusion
- f is lipschiz in D
- use definition of convexity\, can change the domain to [0\,1]\, then apply MVT(inequality) for R -> R^n
inverse function theorem
condition
- Df(x) invertible at a
- f is C1
conclusion
- there is an open ball U centered at a s.t. f is bijective on U and f^-1(x):= g(x) is C1 on f(U)
- can apply chain rule to get Dg(f(a))
steps
- 1. construct a radius and a certain function (involving fixed point) s.t. the function is a contraction on the ball of the radius(bijectivity of U )
- 2. to show f(U) is open. still need the fixed point function
- 3. differentiability of f -> differentiability of g
4. continuity of Df + bijectivity: continuity of Dg
- shaky
implicit function theorem
condition
- f(a\,b) =0
- Dxf(a\,b) invertible
conclusion
- 1. there exist an open set W\, a function g s.t. all y in W\, there is a unique x s.t g(y) =x and f(x\,y)=0
- 2. g is C1
- Dg(b) = -Dyf(a\,b)(Dxf(a\,b))^-1
- key: complete a new function to make it square\!
- if f(a\,b)=0 and Dxf(a\,b) invertible\, then x can be solved uniquely around y=b with small enough r/ y can be solved around x=a if Dy(a\,b) invertible
differentiation across integral sign
condition
- the partial derivative is uniformly continuous on an interval(including s)
- D2f(x\,s) exist for all x
conclusion
a new uniform continuity
- continuous-> discrete -> continuous
- interchanging of integral and differentiating sign
higher order derivative
condition
- all second partial derivatives are continuous\, i\,e. f is C2
conclusion
- D12f = D21 f
key: double MVT
- two differences + order of limit
quadratic approximation for Rn to R function
- condition: C2 function
- conclusion: existence of remaining quadratic form
application: second derivative test
- supportive definition: definite positive/negative/ indefinite
lagrange multiplier method
- condition: two C1 functions(constraint\, target)\, existence of constrained extrema
- conclusion: parallel relationship of derivative at a
- tool: use implicit function theorem with constraint function
integration of functions of several variables
jordan measure
- def
zero measure theorem
- equivalent condition for zero measure domain
- continuous function over zero measure boundary domain is integrable
differentiable surface
- image of C1 function from small space's compact subset to large space
- differentiable surface has zero measure boundary
fubini's theorem
condition
- rectangle
- riemann integrable
- partially integrable
conclusion
- double integral equals riemann integration
- change of integrating order
- double integral in general domain
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