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Isometric embedding
A function g : (Y, p) → (X, d) that preserves the metric, meaning p(y, z) = d(g(y), g(z))

Isometry
 A surjective isometric embedding.
 A function g : (Y, p) → (X, d) that preserves the metric, meaning p(y, z) = d(g(y), g(z))

Converging sequence (x_{n}) ⊆ X, with x_{n}→ x ∈ X
d(x, x_{n}) → 0

Projection to the ith coordinate
 π_{i} : R^{n} → R
 π_{i}(x_{1}, x_{2}, . . . x_{i}, . . . x_{n}) = x_{i}

Cauchy
For every ε > 0, there exists N ∈ N such that for all n, m > N, d(x_{n}, x_{m}) < ε

Complete
Every Cauchy sequence converges

Banach space
A complete normed linear space

Hilbert space
Complete inner product space

Lipschitz function with constant c ≥ 0
d(f(y), f(z)) ≤ cp(y, z) for all y, z ∈ Y

Contraction
Lipschitz function with constant c < 1

Open metric ball about p of radius r in X
B(p, r) = {x ∈ X : d(x, p) < r}

Bounded set A ∈ X
For some x ∈ X and some r > 0, A ⊆ B(x, r)

Fixed point of f : X → X
F(x) = x

Fixed point property
Any continuous function mapping that set to itself has a fixed point

Interior of a set A ⊆ X
 Int A = {x ∈ X : ∃r > 0 with B(x, r) ⊆ A}
 Really inside A (have some metric ball contained in A)

Exterior of a set A ⊆ X
 Ext A = {x ∈ X : ∃r > 0 with B(x, r) ∩ A = ∅}
 Really outside A (have some metric ball which misses A entirely)

Boundary of a set A ⊆ X
 ∂A = {x ∈ X : ∀r > 0, B(x, r) ∩ A ≠ ∅ and B(x, r) − A ≠ ∅}
 Right on the edge (every metric ball is part inside and part outside)

Open subset U ⊆ X
 U = Int U
 Every point is an interior point

Closed subset C ⊆ X
 ∂C ⊆ C
 Contains all its boundary points

Limit point b of C ⊆ X
 ∃ (c_{n}) ⊆ C − {b} with c_{n} → b
 There is a sequence that converges to b

Closure of A ⊆ X
Ā = A ∪ ∂A

A function ƒ is continuous at b ∈ X
For any sequence (x_{n}) ⊆ X with x_{n} → b, ƒ(x_{n}) → f(b)

A function ƒ is continuous
ƒ is continuous at each point of X

Coordinate functions for a function F : X → R^{m}
ƒ_{1}, . . . ƒ_{m} : X → R defined by ƒ_{i} = π_{i}∘F

Open function ƒ : X → Y
 ∀ U open in X, ƒ(U) is open in Y
 Maps open sets to open sets

Homeomorphism
 A continuous, bijective function ƒ : X → Y such that ƒ^{1} : Y → X is also continuous
 ƒ preserves any properties which can be defined using open sets, such as convergence of sequences
 Every isometry is a homeomorphism, but not visa versa

Embedding
ƒ : X → Y such that ƒ : X → ƒ(X) is a homeomorphism, where ƒ(X) is a subspace of Y

Compact
X such that every sequence has a subsequence which converges to a point of X

Distance between A, B ⊆ X
d(A, B) = inf d(a, b) where a ∈ A, b ∈ B

Maximizer of ƒ : A → R
a ∈ A such that ƒ(a) ≥ ƒ(x) for all x ∈ A

Strong maximizer of ƒ : A → R
a ∈ A such that ƒ(a) > ƒ(x) for all x ∈ A  {a}

Minimizer of ƒ : A → R
b ∈ A such that ƒ(b) ≤ ƒ(x) for all x ∈ A

Strong minimizer of ƒ : A → R
b ∈ A such that ƒ(b) < ƒ(x) for all x ∈ A  {b}

Extreme of ƒ : A → R
A minimizer or maximizer of ƒ

Uniformly continuous
A function ƒ : X → Y such that for every ε > 0, there exists δ > 0 such that d(ƒ(u), ƒ(v)) < ε for any u, v ∈ X with d(u, v) < δ

Separation of X
(U, V), where U, V are open nonempty subsets of X such that X = U ∪ V and U ∩ V = ∅

Disconnected
A metric space that has a separation (can be written as the union of nonempty open sets)

Connected
A metric space that is not disconnected (cannot be written as the union of nonempty open sets)

Path in X
 A continuous function γ : [a, b] → X, where a, b ∈ R and a < b
 A path from γ(a) to γ(b)

Path connected
A metric space X such that for each x, y ∈ X, there is a path from x to y in X

Convex
A subset A of a normed linear space V such that for any v, u ∈ A, the line segment from u to v is in A

lim_{x→b} ƒ(x) = L
For every sequence (x_{n}) ⊆ X − {b} with x_{n} → b, ƒ(x_{n}) → L, where b is a limit point of X

ƒ approximates g to nth order at b
g(b) = ƒ(b) and lim_{x→b} d(ƒ(x), g(x))∕[d(x, b)]^{n} = 0

Linear function
 L : V → W for V, W real vector spaces, such that L(αv) = αL(v) and L(v + u) = L(v) + L(u) for any v, u ∈ V and α ∈ R
 Preserves addition and scalar multiplication

Affine function
 T : V → W defined by T(x) = L(x) + c, where L is linear and c ∈ W
 Constant distance from a linear function; only for real vector spaces

Differentiable at b
F : U → R^{m} where U is open in R^{n}, such that there is an affine function T_{b} which approximates F to 1order at b

Differentiable
A function that is differentiable at all points in the domain

R^{n × m}
The collection of n × m matrices with real entries

Norm on R^{n × m}
A = √(∑_{i=1}^{n} ∑_{j=1}^{m} a_{i, j}^{2})


C^{n}(U, R)
{ƒ : U → R : all partials of degree at most n exist and are continuous} where U is open in R^{n}

ƒ ∈ C^{∞}(U, R)
A function that is C^{n} for all n

Directional derivative
 For ƒ : U → R at b ∈ U (an open subset of R^{n}) with respect to u ∈ R^{n}

Local maximizer of ƒ : X → R
p ∈ X such that there is an open U∋p such that ƒ(p) ≥ ƒ(x) for all x ∈ U

Strong local maximizer of ƒ : X → R
p ∈ X such that there is an open U∋p such that ƒ(p) > ƒ(x) for all x ∈ U − {p}

Local minimizer of ƒ : X → R
p ∈ X such that there is an open U∋p such that ƒ(p) ≤ ƒ(x) for all x ∈ U

Strong local minimizer of ƒ : X → R
p ∈ X such that there is an open U∋p such that ƒ(p) < ƒ(x) for all x ∈ U − {p}

Local extreme of ƒ : X → R
A local minimizer or maximizer of ƒ

dx_{i}(v)
v_{i}, where v = (v_{1} . . . v_{n}) an nvector

Differential 0form on A ⊆ R^{n}
A function ƒ : A → R

Differential 1form ω on A ⊆ R^{n}
 ω : A × R^{n} → R
 Given coefficient functions f_{i} : A → R for 1 ≤ i ≤ n, ω = ƒ_{1}dx_{1} + . . . + ƒ_{n}dx_{n}

Orientation on P, a polygon in R^{2}
A consistent choice of direction on the boundary of P (CCW is positive, CW is negative)

Oriented area of P
Normal area of P times the sign of the orientation

Oriented polygon in R^{n}
T(P), where T : R^{2} → R^{n} is affine, and P is a polygon in R^{2}

dx_{i} ∧ dx_{j}(P) for P an oriented polygon in R^{n}
The oriented area of the projection Q of P into the x_{i}x_{j}plane

Oriented area of a parallelogram P spanned by u, v nvectors


Oriented rvolume of the projection Q of the parallelepiped spanned by u_{1}, . . . u_{r} in the x_{i1}, . . . x_{ir}hyperplane




ω a C^{n} form
Each coefficient function of ω is C^{n}

Exterior derivative of a differentiable 0form ƒ : U → R

Jacobian matrix of F at p ∈ U
 For F : U → R^{m} where U is open in R^{n}, with component functions ƒ_{i}
 aka total derivative

Total derivative of F at p ∈ U
 For F : U → R^{m} where U is open in R^{n}, with component functions ƒ_{i}
 aka jacobian matrix

Directional derivative of F at p ∈ U with respect to v ∈ R^{n}
 For F : U → R^{m} where U is open in R^{n}

Smooth at p ∈ U
 F : U → R^{m} where U is open in R^{n}, such that DF(p) has rank n
 Total derivative has rank (dimension of row space) equal to number of rows

clevel hypersurface
S_{c} = {x ∈ U : ƒ(x) = c} for ƒ : U → R where U is open in R^{n}, and c ∈ R

itensor on a vector space V over the real numbers
 A function Γ : V^{i} → R that is linear in each coordinate, i.e. for each 1 ≤ k ≤ i,

R^{n1× . . . × ni}
The set of idimensional rectangles of numbers with side lengths n_{k} in the kth direction

A for A ∈ R^{n1× . . . × ni}
 The square root of the sum of the squares of the entries of A
 Isometric to R^{n1 . . . ni}, so it's a norm


mth Taylor polynomial to ƒ at b ∈ U
 For ƒ ∈ C^{m}(U, R) where U is open in R^{n},

Positive definite tensor
An itensor Γ on R^{n} such that for any nonzero u ∈ R^{n}, Γ(u, . . . u) > 0

Negative definite tensor
An itensor Γ on R^{n} such that for any nonzero u ∈ R^{n}, Γ(u, . . . u) < 0

Indefinite tensor
An itensor Γ on R^{n} such that for there exist u, v ∈ R^{n} with Γ(u, . . . u) < 0 and Γ(v, . . . v) > 0

Positive definite matrix
A ∈ R^{n × . . . × n} such that Γ_{A} is positive definite

Negative definite matrix
A ∈ R^{n × . . . × n} such that Γ_{A} is negative definite

Indefinite matrix
A ∈ R^{n × . . . × n} such that Γ_{A} is indefinite

Interval of R^{n}
I = I_{1} × . . . × I_{n}, where each I_{i} is an interval in R

Closed interval of R^{n}
Interval made up of only closed intervals

Open interval of R^{n}
Interval made up of only open intervals

Volume of the interval I = I_{1} × . . . × I_{n}
vol(I) = l(I_{1}) . . . l(I_{n}), the product of the lengths of the intervals

Partition of I = I_{1} × . . . × I_{n}
Δ = Δ_{1} × . . . × Δ_{n}, where each Δ_{i} is a partition of I_{i}, a closed interval of R

Subinterval J of Δ a partition of I
 The product of one subinterval from each of the partitions that make up Δ
 Write J ∈ Δ

Maximum on a subinterval J ∈ Δ
M_{J} = sup ƒ_{J}, where Δ is a partition of I, a closed interval of R^{n}, and ƒ : I → R a bounded function

Minimum on a subinterval J ∈ Δ
m_{J} = inf ƒ_{J}, where Δ is a partition of I, a closed interval of R^{n}, and ƒ : I → R a bounded function

Upper sum of ƒ for the partition Δ

Lower sum of ƒ for the partition Δ

Refinement of the partition Δ = Δ_{1} × . . . × Δ_{n}
Π = Π_{1} × . . . × Π_{n}, where each Π_{i} is a refinement of Δ_{i} (meaning Δ_{i} ⊆ Π_{i})

Common refinement of the partitions Δ = Δ_{1} × . . . × Δ_{n} and Π = Π_{1} × . . . × Π_{n}
[Δ_{1} ∪ Π_{1}] × . . . × [Δ_{n} ∪ Π_{n}]

A function ƒ : I → R is integrable, where I is a closed interval of R^{n} and ƒ is bounded
sup{L(ƒ, Δ) : Δ is a partition of I} = inf{U(ƒ, Π) : Π is a partition of I}

Integral of an integrable function ƒ

Diameter of A
 For X a metric space, and A ⊆ X,


ncube
A closed interval I = I_{1} × . . . × I_{n} of R^{n} with the property that l(I_{1}) = . . . = l(I_{n})

Cubical partition of Δ
Every subinterval of Δ is an ncube;

Extension by zero of ƒ to B
 For A ⊆ B ⊆ R^{n} and ƒ : A → R, g : B → R is the extension, defined by

Characteristic function of the set A ⊆ R^{n} with A bounded

A ⊆ R^{n} has welldefined volume
 χ_{A} is integrable; define

Translate of A ⊆ R^{n} by u ∈ R^{n}
A + u = {a + u : a ∈ A}

Jordan domain
A bounded subset D of R^{n} such that vol ∂D = 0

Parallelepiped based at p ∈ R^{n} spanned by nvectors u_{1}, . . . u_{m}

Rotation by θ ∈ R
 R_{θ} : R^{2} → R^{2} the linear function with matrix

Hyperplane in R^{n}
 A translation of a n − 1 dimensional sub(vector) space of R^{n}
 The solution sets of a single linear equation

Shear in the ith coordinate in R^{n}
 Linear function L : R^{n} → R^{n} such that L(e_{j}) = e_{j} for j ≠ i, and L(e_{i}) • e_{i} = 1. Shear in the first coordinate is represented by


Jacobian of F : U → R^{n} differentiable at u ∈ U, with U open in R^{n}
J_{F}(u) = det DF(u)

