Change of variables formula (one standard Riemann form): Let U,V⊆RnU,V\subseteq\mathbb{R}^nU,V⊆Rn be open and let Φ:U→V\Phi:U\to VΦ:U→V be a C1C^1C1 diffeomorphism . Let E⊆UE\subseteq UE⊆U be a set such that EEE and Φ(E)\Phi(E)Φ(E) are “nice” for Riemann integration (e.g., bounded with boundary of measure zero ). If fff is Riemann integrable on Φ(E)\Phi(E)Φ(E), then f∘Φ⋅∣det⁡DΦ∣f\circ \Phi\cdot |\det D\Phi|f∘Φ⋅∣detDΦ∣ is Riemann integrable on EEE and ∫Φ(E)f(x) dx=∫Ef(Φ(u)) ∣det⁡DΦ(u)∣ du. \int_{\Phi(E)} f(x)\,dx = \int_E f(\Phi(u))\,\bigl|\det D\Phi(u)\bigr|\,du. ∫Φ(E)​f(x)dx=∫E​f(Φ(u))​detDΦ(u)​du. ...

Let U,V⊆RnU,V\subseteq\mathbb{R}^nU,V⊆Rn be open and let Φ:U→V\Phi:U\to VΦ:U→V be a C1C^1C1 diffeomorphism . Let E⊆UE\subseteq UE⊆U be a region for which the Riemann integral behaves well (e.g., bounded with boundary of measure zero), and let fff be Riemann integrable on Φ(E)\Phi(E)Φ(E). Corollary: The function u↦f(Φ(u)) ∣det⁡DΦ(u)∣u\mapsto f(\Phi(u))\,|\det D\Phi(u)|u↦f(Φ(u))∣detDΦ(u)∣ is Riemann integrable on EEE, and the integrals are related by ∫Φ(E)f(x) dx=∫Ef(Φ(u)) ∣det⁡DΦ(u)∣ du. \int_{\Phi(E)} f(x)\,dx = \int_E f(\Phi(u))\,\bigl|\det D\Phi(u)\bigr|\,du. ∫Φ(E)​f(x)dx=∫E​f(Φ(u))​detDΦ(u)​du. Connection to parent theorem: This is the change of variables theorem for multiple integrals, often packaged as the practical rule “insert the absolute Jacobian determinant when changing coordinates.” ...

Let XXX be a set and let A⊆XA\subseteq XA⊆X. The characteristic function (or indicator function) of AAA is the function 1A:X→{0,1}\mathbf{1}_A:X\to\{0,1\}1A​:X→{0,1} defined by 1A(x):={1,x∈A,0,x∉A. \mathbf{1}_A(x):= \begin{cases} 1,& x\in A,\\ 0,& x\notin A. \end{cases} 1A​(x):={1,0,​x∈A,x∈/A.​Indicator functions convert set membership questions into algebraic statements and are a standard device in integration and measure theory (e.g., simple functions are finite linear combinations of indicators). ...

Let I⊆RI\subseteq\mathbb{R}I⊆R be an interval and let k∈N∪{0}k\in\mathbb{N}\cup\{0\}k∈N∪{0}. A function f:I→Rf:I\to\mathbb{R}f:I→R (or C\mathbb{C}C) is of class CkC^kCk on III if: f(j)f^{(j)}f(j) exists on III for every integer jjj with 0≤j≤k0\le j\le k0≤j≤k (where f(0):=ff^{(0)}:=ff(0):=f), and each derivative f(j)f^{(j)}f(j) is continuous on III. The class CkC^kCk encodes smoothness needed for Taylor’s theorem, inverse/implicit function statements (in higher dimensions), and many approximation results. ...

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​). For an integer r≥0r\ge 0r≥0, write ∂α\partial^\alpha∂α for partial derivatives corresponding to a multi-index α=(α1,…,αk)∈Nk\alpha=(\alpha_1,\dots,\alpha_k)\in\mathbb{N}^kα=(α1​,…,αk​)∈Nk with order ∣α∣=α1+⋯+αk|\alpha|=\alpha_1+\cdots+\alpha_k∣α∣=α1​+⋯+αk​. ...

Let (X,d)(X,d)(X,d) be a metric space , let x∈Xx\in Xx∈X, and let r≥0r\ge 0r≥0. The closed ball of radius rrr centered at xxx is B‾(x,r):={y∈X:d(x,y)≤r}.\overline{B}(x,r):=\{y\in X : d(x,y)\le r\}.B(x,r):={y∈X:d(x,y)≤r}.Closed balls are typically closed sets in metric spaces and are used to describe boundedness and compactness phenomena (e.g., bounded sets lie inside some closed ball). Compare with open ball . ...

Let (X,d)(X,d)(X,d) be a metric space . A subset F⊆XF\subseteq XF⊆X is closed if its complement X∖FX\setminus FX∖F is open . In metric spaces, closedness is equivalent to several other important properties (e.g., containing limits of convergent sequences from the set). Closed sets are stable under intersections and are used in defining closures and compactness . Examples: In R\mathbb{R}R, the interval [a,b][a,b][a,b] is closed. In R\mathbb{R}R, Z\mathbb{Z}Z is closed (its complement is a union of open intervals). In any metric space, ∅\varnothing∅ and XXX are closed.

Closed sets are complements of open sets: In a metric space (X,d)(X,d)(X,d), a set F⊆XF\subseteq XF⊆X is closed if and only if X∖FX\setminus FX∖F is open . Consequently: arbitrary intersections of closed sets are closed, and finite unions of closed sets are closed. This duality between open and closed sets is a basic tool in topology and analysis, especially for closure , limit points , and compactness arguments. ...

Closed subset of compact is compact: Let (X,d)(X,d)(X,d) be a metric space , let K⊆XK\subseteq XK⊆X be compact , and let F⊆KF\subseteq KF⊆K be closed in XXX (equivalently, closed in the subspace KKK). Then FFF is compact. This permanence property is used constantly: once compactness is established, it automatically applies to all closed substructures. Proof sketch: Let {Uα}\{U_\alpha\}{Uα​} be an open cover of FFF in XXX. Then {Uα}∪{X∖F}\{U_\alpha\}\cup\{X\setminus F\}{Uα​}∪{X∖F} is an open cover of KKK. By compactness of KKK, there is a finite subcover. Removing X∖FX\setminus FX∖F leaves a finite subcover of FFF. ...

Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. The closure of AAA, denoted A‾\overline{A}A (or cl⁡(A)\operatorname{cl}(A)cl(A)), is the set A‾:={x∈X:∀r>0, B(x,r)∩A≠∅}. \overline{A}:=\{x\in X : \forall r>0,\ B(x,r)\cap A\neq\varnothing\}. A:={x∈X:∀r>0, B(x,r)∩A=∅}. Equivalently, A‾\overline{A}A is the intersection of all closed sets that contain AAA. The closure adds to AAA all points that can be approximated arbitrarily well by points of AAA. It is fundamental for density , limit points , and topological convergence. ...