Inverse function

Let f:X→Yf:X\to Yf:X→Y be bijective . The inverse function of fff is the function f−1:Y→Xf^{-1}:Y\to Xf−1:Y→X defined by f−1(y)=xwhere x∈X is the unique element with f(x)=y.f^{-1}(y)=x\quad\text{where $x\in X$ is the unique element with }f(x)=y.f−1(y)=xwhere x∈X is the unique element with f(x)=y. It satisfies ...

Inverse Function Theorem (multivariable)

Inverse Function Theorem: Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn be open and let f:U→Rnf:U\to\mathbb{R}^nf:U→Rn be of class C1C^1C1. Suppose a∈Ua\in Ua∈U and det⁡Df(a)≠0. \det Df(a)\neq 0. detDf(a)=0. Let b=f(a)b=f(a)b=f(a). Then there exist open neighborhoods U0U_0U0 of aaa and V0V_0V0 of bbb such that: ...

Inverse Function Theorem (one variable)

Inverse Function Theorem (one variable): Let I⊆RI\subseteq\mathbb{R}I⊆R be an interval and let f:I→Rf:I\to\mathbb{R}f:I→R be continuous and strictly monotone . Then fff is a bijection from III onto J=f(I)J=f(I)J=f(I), so the inverse f−1:J→If^{-1}:J\to If−1:J→I exists and is continuous. If moreover x0∈I∘x_0\in I^\circx0∈I∘ and fff is differentiable at x0x_0x0 with f′(x0)≠0f'(x_0)\neq 0f′(x0)=0, then f−1f^{-1}f−1 is differentiable at y0=f(x0)y_0=f(x_0)y0=f(x0) and (f−1)′(y0)=1f′(x0). (f^{-1})'(y_0)=\frac{1}{f'(x_0)}. (f−1)′(y0)=f′(x0)1. In particular, if f∈C1(I∘)f\in C^1(I^\circ)f∈C1(I∘) and f′(x)≠0f'(x)\neq 0f′(x)=0 for all x∈I∘x\in I^\circx∈I∘, then f−1∈C1(J∘)f^{-1}\in C^1(J^\circ)f−1∈C1(J∘) and (f−1)′(y)=1f′(f−1(y))(y∈J∘). (f^{-1})'(y)=\frac{1}{f'(f^{-1}(y))}\qquad (y\in J^\circ). (f−1)′(y)=f′(f−1(y))1(y∈J∘). ...

Isolated point

Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. A point x∈Ax\in Ax∈A is an isolated point of AAA if there exists r>0r>0r>0 such that B(x,r)∩A={x}.B(x,r)\cap A=\{x\}.B(x,r)∩A={x}.Isolated points are the opposite of limit points: near an isolated point there are no other points of the set. Sets can have both isolated points and limit points. Examples: ...

Isometry

Let (X,dX)(X,d_X)(X,dX) and (Y,dY)(Y,d_Y)(Y,dY) be metric spaces . A function f:X→Yf:X\to Yf:X→Y is an isometry if ∀x1,x2∈X,dY(f(x1),f(x2))=dX(x1,x2).\forall x_1,x_2\in X,\quad d_Y\bigl(f(x_1),f(x_2)\bigr)=d_X(x_1,x_2).∀x1,x2∈X,dY(f(x1),f(x2))=dX(x1,x2).Isometries preserve all metric structure: convergence , Cauchy sequences , completeness , and (when fff is bijective onto its image) the induced topology. They are the natural notion of “rigid motion” in metric spaces. ...

Iterated integral

Let R=[a,b]×[c,d]⊆R2R=[a,b]\times[c,d]\subseteq \mathbb{R}^2R=[a,b]×[c,d]⊆R2 and let f:R→Rf:R\to\mathbb{R}f:R→R be a function such that for each fixed x∈[a,b]x\in[a,b]x∈[a,b] the function y↦f(x,y)y\mapsto f(x,y)y↦f(x,y) is (Riemann) integrable on [c,d][c,d][c,d], and similarly for each fixed yyy. The iterated integrals are ∫ab(∫cdf(x,y) dy)dxand∫cd(∫abf(x,y) dx)dy,\int_a^b\left(\int_c^d f(x,y)\,dy\right)dx \quad\text{and}\quad \int_c^d\left(\int_a^b f(x,y)\,dx\right)dy,∫ab(∫cdf(x,y)dy)dxand∫cd(∫abf(x,y)dx)dy, provided the inner integrals exist and the resulting outer integrals exist. ...

Jacobian determinant

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open and let f:U→Rkf:U\to\mathbb{R}^kf:U→Rk. If fff has a Jacobian matrix Jf(a)J_f(a)Jf(a) at a∈Ua\in Ua∈U, the Jacobian determinant of fff at aaa is det⁡Jf(a)∈R.\det J_f(a)\in\mathbb{R}.detJf(a)∈R.When fff is $C^1$ , det⁡Jf(a)≠0\det J_f(a)\neq 0detJf(a)=0 is the nondegeneracy condition in the inverse function theorem. In integration, ∣det⁡Jf∣|\det J_f|∣detJf∣ appears in the change-of-variables formula as the local volume scaling factor. ...

Jacobian matrix

Let U⊆RkU\subseteq\mathbb{R}^kU⊆Rk be open and let f:U→Rmf:U\to\mathbb{R}^mf:U→Rm with components f=(f1,…,fm)f=(f_1,\dots,f_m)f=(f1,…,fm), where each fi:U→Rf_i:U\to\mathbb{R}fi:U→R. If all relevant partial derivatives exist at a∈Ua\in Ua∈U, the Jacobian matrix of fff at aaa is the m×km\times km×k matrix ...

Jordan content

A (closed) rectangle in Rk\mathbb{R}^kRk is a set of the form R=∏j=1k[aj,bj],aj≤bj,R=\prod_{j=1}^k [a_j,b_j],\qquad a_j\le b_j,R=j=1∏k[aj,bj],aj≤bj, with volume vol⁡(R)=∏j=1k(bj−aj).\operatorname{vol}(R)=\prod_{j=1}^k (b_j-a_j).vol(R)=j=1∏k(bj−aj).An elementary set is a finite union of rectangles. If an elementary set EEE is written as a finite union of pairwise disjoint rectangles R1,…,RNR_1,\dots,R_NR1,…,RN, its Jordan content (also called content) is ...

Jordan decomposition lemma (bounded variation)

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be a function of bounded variation . For x∈[a,b]x\in[a,b]x∈[a,b], write Vax(f)=sup⁡{∑j=1n∣f(tj)−f(tj−1)∣: a=t0<t1<⋯<tn=x} V_a^x(f)=\sup\left\{\sum_{j=1}^n |f(t_j)-f(t_{j-1})|:\ a=t_0<t_1<\cdots<t_n=x\right\} Vax(f)=sup{∑j=1n∣f(tj)−f(tj−1)∣: a=t0<t1<⋯<tn=x} for the total variation of fff on [a,x][a,x][a,x]. ...