nyuuzyou/wikis huggingface dataset:
Topology Defined by Basis

== Theorem ==
Let $S$ be a set.
Let $\BB$ be a set of subsets of $S$.
Suppose that
:$(\text B1): \quad \forall A_1, A_2 \in \BB: \forall x \in A_1 \cap A_2: \exists A \in \BB: x \in A \subseteq A_1 \cap A_2$
:$(\text B2): \quad \forall x \in X: \exists A \in \BB: x \in A$
::$\tau = \set {\bigcup \GG: \GG \subseteq \BB}$
Then:
:$T = \struct {S, \tau}$ is a topological space
:$\BB$ is a basis of $T$.
== Proof ==
We have to prove Open Set Axioms:
=== ===
Let $\FF \subseteq \tau$.
Define by definition of $\tau$ a family $\family {\GG_A}_{A \mathop \in \FF}$ such that
:$\forall A \in \FF: A = \bigcup \GG_A \land \GG_A \subseteq \BB$.
By General Self-Distributivity of Set Union:
:$\ds \bigcup \bigcup_{A \mathop \in \FF} \GG_A = \bigcup_{A \mathop \in \FF} \bigcup \GG_A = \bigcup \FF$
By Union of Family of Subsets is Subset:
:$\ds \bigcup_{A \mathop \in \FF} \GG_A \subseteq \BB$
Thus by definition of $\tau$
:$\bigcup \FF \in \tau$
=== ===
Let $A$ and $B$ be elements of $\tau$.
By definition of $\tau$ there exist subsets $\GG_A$ and $\GG_B$ of $\BB$ such that:
:$A = \bigcup \GG_A$ and $B = \bigcup \GG_B$ and $\GG_A, \GG_B \subseteq \BB$
Set $\GG_C = \set {C \in \BB: C \subseteq A \cap B}$
By Union of Subsets is Subset:
:$\bigcup \GG_C \subseteq A \cap B$
We will prove the inclusion: $A \cap B \subseteq \bigcup \GG_C$
Let $x \in A \cap B$.
Then by definition of intersection:
:$v \in A$ and $x \in B$
Hence by definition of union:
:$\exists D \in \GG_A: x \in D$
Analogically:
:$\exists E \in \GG_B: x \in E$
By definition of subset $D, E \in \BB$ and by definition of intersection $x \in D \cap E$.
Then by $(\text B 1)$:
:$\exists U \in \BB: x \in U \subseteq D \cap E$
By Set is Subset of Union/Set of Sets:
:$D \subseteq A$ and $E \subseteq B$
Then by Set Intersection Preserves Subsets:
:$D \cap E \subseteq A \cap B$
Hence by Subset Relation is Transitive:
:$U \subseteq A \cap B$
Then by definition of $\GG_C$:
:$U \in \GG_C$
Thus by definition of union:
:$x \in \bigcup \GG_C$
This ends the proof of inclusion.
Then by definition of set equality:
:$A \cap B = \bigcup \GG_C$
By definition of subset:
:$\GG_C \subseteq \BB$
Thus by definition of $\tau$:
:$A \cap B \in \tau$
=== ===
By $(\text B 2)$ and definition of union:
:$\bigcup \BB = X$
Because $\BB \subseteq \BB$ by definition of $\tau$:
:$S \in \tau$
It remains to prove that $\BB$ is a basis of $T$.
Let $U$ be an open set of $S$.
Let $x$ be a point of $S$ such that $x \in U$
By definition of $\tau$ there exists $\GG \subseteq \BB$ such that:
:$U = \bigcup \GG$
By definition of union:
:$\exists A \in \GG: x \in A$
By definition of subset:
$A \in \BB$
Thus by Set is Subset of Union:
:$x \in A \subseteq U$
All the open set axioms are fulfilled, and the result follows by definition of basis.
== Sources ==
* 
* 
Category:Topological Bases
Topology Defined by Closed Sets

== Theorem ==
Let $S$ be a set.
Let $\tau$ be a set of subsets of $S$.
Then $\tau$ is a topology on $S$ :
:$(1): \quad$ Any intersection of arbitrarily many closed sets of $S$ under $\tau$ is a closed set of $S$ under $\tau$
:$(2): \quad$ The union of any finite number of closed sets of $S$ under $\tau$ is a closed set of $S$ under $\tau$
:$(3): \quad S$ and $\O$ are both closed sets of $S$ under $\tau$
where a closed set $V$ of $S$ under $\tau$ is defined as a subset of $S$ such that $S \setminus V \in \tau$.
== Proof ==
From the definition, if $V$ is a closed set of $S$, then $S \setminus V$ is an open set of $S$.
Let $\mathbb V$ be any arbitrary set of closed sets of $S$.
Then by De Morgan's Laws: Difference with Intersection, we have:
:$\ds S \setminus \bigcap \mathbb V = \bigcup_{V \mathop \in \mathbb V} \paren {S \setminus V}$
First, let $\tau$ be a topology on $S$.
We have that:
:Intersection of Closed Sets is Closed in Topological Space
:Finite Union of Closed Sets is Closed in Topological Space
:By Open and Closed Sets in Topological Space, $\O$ and $S$ are both closed in $S$.
Thus, the properties as listed above hold.
Suppose the properties:
:$(1): \quad$ Any intersection of arbitrarily many closed sets of $S$ under $\tau$ is a closed set of $S$ under $\tau$
:$(2): \quad$ The union of any finite number of closed sets of $S$ under $\tau$ is a closed set of $S$ under $\tau$
:$(3): \quad S$ and $\O$ are both closed sets of $S$ under $\tau$.
all hold.
That means $\ds \bigcap \mathbb V$ is closed.
So $\ds S \setminus \bigcap \mathbb V = \bigcup_{V \mathop \in \mathbb V} \paren {S \setminus V}$ is open.
Thus we have that the union of arbitrarily many open sets of $S$ under $\tau$ is an open set of $S$ under $\tau$.
Similarly, we deduce that the intersection of any finite number of open sets of $S$ under $\tau$ is an open set of $S$ under $\tau$.
By Open and Closed Sets in Topological Space, $\O$ and $S$ are both open in $S$.
So $\tau$ is a topology on $S$.
== Sources ==
* : Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets: Theorem $2$
* : $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Proposition $3.7.4$
* : Part $\text I$: Basic Definitions: Section $1$: General Introduction
Category:Topology
Category:Closed Sets
Topology Defined by Neighborhood System

== Theorem ==
Let $S$ be a set.
Let $\family {\NN_x}_{x \mathop \in S}$ be an indexed family where $\NN_x$ is non-empty set of subsets of $S$.
Assume that
:$(\text N 1): \quad \forall x \in S, U \in \NN_x: x \in U$
:$(\text N 2): \quad \forall x \in S, U \in \NN_x, y \in U:\exists V \in \NN_y: V \subseteq U$
:$(\text N 3): \quad \forall x \in S, U_1, U_2 \in \NN_x: \exists U \in \NN_x: U \subseteq U_1 \cap U_2$
Then $T = \struct {S, \tau}$ is a topological space where:
:$\ds \tau = \set {\bigcup \GG: \GG \subseteq \bigcup_{x \mathop \in S} \NN_x}$
Moreover, $\family {\NN_x}_{x \mathop \in S}$ is a neighborhood system of $T$.
== Proof ==
Define:
:$\BB := \ds \bigcup_{x \mathop \in S} \NN_x$
According to Topology Defined by Basis it should be proved that
:$(\text B 1): \quad \forall A_1, A_2 \in \BB: \forall x \in A_1 \cap A_2: \exists A \in \BB: x \in A \subseteq A_1 \cap A_2$
:$(\text B 2): \quad \forall x \in S: \exists A \in \BB: x \in A$
Ad $(\text B 1)$:
Let $A_1, A_2 \in \BB$.
Let $x \in A_1 \cap A_2$.
By definition of intersection:
:$x \in A_1 \land x \in A_2$
By definition of union:
:$\exists x_1 \in S: A_1 \in \NN_{x_1}$
and
:$\exists x_2 \in S: A_2 \in \NN_{x_2}$
By $(\text N 2)$:
:$\exists V_1 \in \NN_x: V_1 \subseteq A_1$
and
:$\exists V_2 \in \NN_x: V_2 \subseteq A_2$
By $(\text N 3)$:
:$\exists U \in \NN_x: U \subseteq V_1 \cap V_2$
By definition of union:
:$U \in \BB$
By Set Intersection Preserves Subsets:
:$V_1 \cap V_2 \subseteq A_1 \cap A_2$
Thus by Subset Relation is Transitive:
:$U \subseteq A_1 \cap A_2$
Ad $(\text B 2)$:
Let $x \in S$.
By definition of empty set:
:$\exists U: U \in \NN_x$
By $(\text N 1)$:
:$x \in U$
By definition of union:
:$U \in \BB$
Thus:
:$\exists A \in \BB: x \in A$
By definition of $\tau$ and $\BB$:
:$\tau = \set {\bigcup \GG: \GG \subseteq \BB}$
Thus by Topology Defined by Basis:
:$T = \struct {S, \tau}$ is a topological space
It remains to prove that
:$\family {\NN_x}_{x \mathop \in S}$ is a neighborhood system
Let $x \in S$.
Let $U \in \tau$.
By definition of $\tau$:
:$\exists \GG \subseteq \BB: U = \bigcup \GG$
Let $x \in U$.
By definition of union:
:$\exists V \in \GG: x \in V$
By definition of subset:
:$V \in \BB$
By definition of union:
:$\exists y \in S: V \in \NN_y$
By $(\text N 2)$:
:$\exists W \in \NN_x: W \subseteq V$
By Set is Subset of Union/Set of Sets:
:$V \subseteq U$
Thus by Subset Relation is Transitive:
:$\exists W \in \NN_x: W \subseteq U$
Thus by definition:
:$\NN_x$ is a local basis.
== Sources ==
* 
* 
Category:Topology
Topology Generated by Closed Sets

== Theorem ==
Let $X$ be a set.
Let $\FF$ be a set of subsets of $X$ such that:
:$\O \in \FF$
:$\forall A, B \in \FF: A \cup B \in \FF$
:$\forall \GG \subseteq \FF: \bigcap \GG \in \FF$
Let $\tau = \set {\relcomp X A: A \in \FF}$.
Then:
:$T = \struct {X, \tau}$ is topological space and
:for every subset $A$ of $X$, $A$ is closed in $T$ $A \in \FF$.
== Proof ==
== Sources ==
* 
* 
Category:Topology
Topology as Magma of Sets

== Theorem ==
The concept of a topology is an instance of a magma of sets.
== Proof ==
It will suffice to define partial mappings such that the axiom for a magma of sets crystallises into the axioms for a topology.
Let $X$ be any set, and let $\powerset X$ be its power set.
Define:
:$\phi_1: \powerset X \to \powerset X: \map {\phi_1} S := X$
:$\phi_2: \powerset X^2 \to \powerset X: \map {\phi_2} {S, T} := S \cap T$
For each index set $I$, define:
:$\ds \phi_I: \powerset X^I \to \powerset X: \map {\phi_I} {\family {S_i}_{i \mathop \in I} } := \bigcup_{i \mathop \in I} S_i$
It is blatantly obvious that these partial mappings capture the axioms for a topology.
Category:Topology
Category:Magmas of Sets
Topology forms Complete Lattice

== Theorem ==
Let $\struct {X, \tau}$ be a topological space.
Then $\struct {\tau, \subseteq}$ is a complete lattice.
== Proof ==
To show that $\struct {\tau, \subseteq}$ is a complete lattice, we must show that every subset of $\tau$ has a supremum and an infimum.
Let $S \subseteq \tau$.
By the definition of a topology:
:$\ds \bigcup S \in \tau$
By Union is Smallest Superset, $\ds \bigcup S$ is the supremum of $S$.
Let $I$ be the interior of $\ds \bigcap S$, where by Intersection of Empty Set $\ds \bigcap \O$ is conventionally taken to be $X$.
Then by the definition of interior and Intersection is Largest Subset:
:$I \in \tau$
and
:$I \subseteq U$
for each $U \in S$.
Let $V \in \tau$ with $V \subseteq U$ for each $U \in S$.
By Intersection is Largest Subset:
:$\ds V \subseteq \bigcap S$.
Then by the definition of interior:
:$V \subseteq I$
Thus $I$ is the infimum of $S$.
So each subset of $\tau$ has a supremum and an infimum.
Thus, by definition, $\struct {\tau, \subseteq}$ is a complete lattice.
Category:Topology
Category:Complete Lattices
Topology induced by Scaled Euclidean Metric on Positive Integers is Discrete

== Theorem ==
Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $\delta: \Z_{>0} \times \Z_{>0} \to \R$ be the metric on $\Z_{>0}$ defined as:
:$\forall x, y \in \Z_{>0}: \map \delta {x, y} = \dfrac {\size {x - y} } {x y}$
Then the metric topology for $\delta$ is a discrete topology.
== Proof ==
Let $\tau_\delta$ denote the metric topology for $\delta$.
In Scaled Euclidean Metric is Metric it is demonstrated that $\delta$ is indeed a metric on $\Z_{>0}$.
Let $a \in \Z_{>0}$.
Recall the definition of the open $\epsilon$-ball of $a$ in $\struct {\Z_{>0}, \delta}$:
:$\map {B_\epsilon} a := \set {x \in A: \map \delta {x, a} < \epsilon}$
Let $x \in \R_{>0}$.
Let $\epsilon \in \R_{>0}$ such that $\epsilon < \dfrac 1 {a \paren {a + 1} }$.
But we have:
and so:
:$\forall x \in \Z_{>0}, x \ne a: x \notin \map {B_\epsilon} a$
It follows that:
:$\map {B_\epsilon} a := \set a$
Thus by definition of $\tau_d$:
:$\forall a \in \Z_{>0}: \set a \in \tau_\delta$
It follows from Basis for Discrete Topology that $\struct {\Z_{>0}, \tau_\delta}$ is a discrete topological space.
== Sources ==
* : Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $10$
Category:Discrete Topology
Category:Metric Spaces
Topology induced by Usual Metric on Positive Integers is Discrete

== Theorem ==
Let $\Z_{>0}$ be the set of (strictly) positive integers.
Let $d: \Z_{>0} \times \Z_{>0} \to \R$ be the usual (Euclidean) metric on $\Z_{>0}$.
Then the metric topology for $d$ is a discrete topology.
== Proof ==
Let $\tau_d$ denote the metric topology for $d$.
Let $\epsilon \in \R_{>0}$ such that $\epsilon < 1$.
Let $a \in \Z_{>0}$.
Recall the definition of the open $\epsilon$-ball of $a$ in $\struct {\Z_{>0}, d}$:
:$\map {B_\epsilon} a := \set {x \in A: \map d {x, a} < \epsilon}$
But we have:
:$\forall x \in \Z_{>0}, x \ne a: \map d {x, a} \ge 1$
and so:
:$\forall x \in \Z_{>0}, x \ne a: x \notin \map {B_\epsilon} a$
It follows that:
:$\map {B_\epsilon} a := \set a$
Thus by definition of $\tau_d$:
:$\forall a \in \Z_{>0}: \set a \in \tau_d$
It follows from Basis for Discrete Topology that $\struct {\Z_{>0}, \tau_d}$ is a discrete topological space.
== Sources ==
* : Part $\text {II}$: Counterexamples: $32$. Special Subsets of the Real Line: $10$
Category:Discrete Topology
Category:Metric Spaces
Topology is Discrete iff All Singletons are Open

== Theorem ==
Let $\struct {S, \tau}$ be a topological space.
Then:
:$\tau$ is the discrete topology on $S$
:
:$\forall x \in S: \set x \in \tau$
That is, every singleton of $S$ is $\tau$-open.
== Proof ==
=== Sufficient Condition ===
Let $\tau$ be the discrete topology on $S$.
Let $x \in S$ be arbitrary.
Then from Set in Discrete Topology is Clopen it follows directly that $\set x$ is open in $\struct {S, \tau}$.
=== Necessary Condition ===
Let $\struct {S, \tau}$ be such that:
:$\forall x \in S: \set x \in \tau$
Let $T \subseteq S$ be arbitrary.
Then $T = \bigcup \set {\set t: t \in T}$.
We have that each $\set t$ is open in $\struct {S, \tau}$.
By definition of a topology, a union of open sets is open.
Hence $T$ is open in $\struct {S, \tau}$.
As $T$ is arbitrary, it follows by definition that $\tau$ is the discrete topology.
Category:Discrete Topology
Category:Singletons
Topology is Locally Compact iff Ordered Set of Topology is Continuous

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
Let $L = \struct {\tau, \preceq}$ be the ordered set where $\preceq$ is the subset relation.
Then
:$(1): \quad T$ is locally compact implies $L$ is continuous
:$(2): \quad T$ is $T_3$ space and $L$ is continuous implies $T$ is locally compact
== Proof ==
=== Condition $(1)$ ===
Let $T$ be locally compact.
By Topology forms Complete Lattice:
:$L$ is complete lattice.
Thus by Way Below Closure is Directed in Bounded Below Join Semilattice:
:$\forall x \in \tau: x^\ll$ is directed
Thus by definition:
:$L$ is up-complete.
Let $x \in \tau$.
We will prove that
:$x \subseteq \map \sup {x^\ll}$
Let $a \in x$.
By definition:
:$x$ is open set in $T$
By definition of locally compact:
:there exists local basis $\BB$ of $a$ such that all elements are compact.
By definition of local basis:
:$\exists y \in \BB: y \subseteq x$
Then
:$y$ is compact.
By Way Below if Between is Compact Set in Ordered Set of Topology:
:$y \ll x$
By definition of way below closure:
:$y \in x^\ll$
By Set is Subset of Union/Set of Sets:
:$\ds y \subseteq \bigcup \paren {x^\ll}$
By definition of subset:
:$\ds a \in \bigcup \paren {x^\ll}$
Thus by proof of Topology forms Complete Lattice:
:$a \in \sup \paren {x^\ll}$
According to definition of set equality it remains to prove that:
:$\map \sup {x^\ll} \subseteq x$
Let $a \in \map \sup {x^\ll}$.
By proof of Topology forms Complete Lattice:
:$\ds a \in \bigcup \paren {x^\ll}$
By definition of union:
:$\exists y \in x^\ll: a \in y$
By definition of way below closure:
:$y \ll x$
By Way Below implies Preceding:
:$y \preceq x$
Then
:$y \subseteq x$
Thus by definition of subset:
:$a \in x$
By definition:
:$L$ satisfies the axiom of approximation.
Thus by definition
:$L$ is continuous.
=== Condition $(2)$ ===
Let $T$ be a $T_3$ space.
Let $L$ be continuous.
Let $x \in S, A \in \tau$ such that:
:$x \in A$
By definition of continuous:
:$L$ satisfies the axiom of approximation.
By definition of union:
:$\exists y \in A^\ll: x \in y$
By definition of way below closure:
:$y \ll A$
By definition of Definition:T3 Space/Definition 2:
:$\exists N_x: \relcomp S {N_x} \in \tau: \exists V \in \tau: x \in V \subseteq N_x \subseteq y$
By definition of interior:
:$x \in N_x^\circ$
where $N_x^\circ$ denotes the interior of $N_x$.
We will prove that $N_x$ is topologically compact.
Let $\FF$ be a set of open subsets of $S$ such that:
:$\FF$ is a cover of $N_x$
Define a set of open subsets of $S$:
:$\FF' := \FF \cup \set {\relcomp S {N_x} }$
Then
:$\ds A \subseteq \bigcup \FF'$
By Way Below in Ordered Set of Topology:
:there exists a finite subset $\GG'$ of $\FF'$: $\ds y \subseteq \bigcup \GG'$
Define a finite subset of $\FF$:
:$\GG := \GG' \setminus \set {\relcomp S {N_x} }$
By definition of union:
:$\ds N_x \subseteq \bigcup \GG$
Thus by definition:
:$N_x$ has finite subcover of $\FF$
Thus by definition:
:$N_x$ is compact.
Thus by Locally Compact iff Open Neighborhood contains Compact Set;
:$T$ is locally compact.
== Sources ==
* 
* 
* 
Category:Continuous Lattices
Category:Topology
Category:Locally Compact Spaces
Torelli's Sum

== Theorem ==
:$\ds \paren {x + y}^{\overline n} = \sum_k \binom n k x \paren {x - k z + 1}^{\overline {k - 1} } \paren {y + k z}^{\overline {n - k} }$
where:
:$\dbinom n k$ denotes a binomial coefficient
:$x^{\overline k}$ denotes $x$ to the $k$ rising.
== Proof ==
From Rising Factorial as Factorial by Binomial Coefficient:
:$\paren {x + y}^{\overline n} = n! \dbinom {x + y + n - 1} n$
Recall Sum over $k$ of $\dbinom {r - t k} k$ by $\dbinom {s - t \paren {n - k} } {n - k}$ by $\dfrac r {r - t k}$:
:$\ds \sum_{k \mathop \ge 0} \binom {r - t k} k \binom {s - t \paren {n - k} } {n - k} \frac r {r - t k} = \binom {r + s - t n} n$
Let the following substitutions be made:
:$r \gets x$
:$t \gets -\paren {1 - z}$
:$s \gets y - 1 + n z$
and so to obtain:
:$\ds \dbinom {x + y + n - 1} n = \sum_k \dbinom {x + \paren {1 - z} k} k \dbinom {y - 1 + n z + \paren {n - k} \paren {1 - z} } {n - k} \dfrac x {x + \paren {1 - z} k}$
Then:
and:
Hence:
== Sources ==
* 
* : $\S 1.2.6$: Binomial Coefficients: Exercise $34$
Category:Rising Factorials
Torque Applied to Shaft of Gyropscope Perpendicular to Axis results in Precession

== Physical Law ==
Let $G$ be a gyroscope.
Let a torque be applied to the shaft of $G$ in order to get it to rotate the shaft on an axis perpendicular to the shaft.
Then the resulting motion of $G$ will be a precession.
== Proof ==
== Sources ==
* 
* 
Category:Precession
Category:Gyroscopes
Torricelli's Law

== Physical Law ==
The speed $v$ of outflow of a fluid through a hole at the bottom of an open tank filled to a depth $h$ equals the speed $v$ that a body would acquire in falling freely from a height $h$.
== Proof ==
== Sources ==
* : $1$: The Nature of Differential Equations: Miscellaneous Problems for Chapter $1$: $7$
* : Chapter $\text {A}.15$: Torricelli ()
Category:Fluid Mechanics
Torus as Surface of Revolution

== Theorem ==
Let $\struct {\R^3, d}$ be the Euclidean space.
Let $S_C \subseteq \R^3$ be the surface of revolution.
Let $C$ be a circle defined by $x^2 + \paren {y - 2}^2 = 1$ in the open upper half-plane.
Let the smooth local parametrization of $C$ be:
:$\map \gamma t = \tuple {\sin t, 2 + \cos t}$
Then the induced metric on $S_C$ is:
:$g = d t^2 + \paren {2 + \cos t}^2 \, d \theta^2$
== Proof ==
We have that:
:$\map {\gamma'} t = \tuple {\cos t, - \sin t}$
Furthermore:
:$\paren {\cos t}^2 + \paren {- \sin t}^2 = 1$
Hence, $\map \gamma t$ is a unit-speed curve.
By :
:$g = d t^2 + \paren {2 + \cos t}^2 \, d \theta^2$
== Sources ==
* : $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics
Category:Surfaces of Revolution
Category:Solid Geometry
Category:Induced Metrics
Total Boundedness is not Preserved under Homeomorphism

== Theorem ==
Let $M = \struct {A, d}$ be a totally bounded metric space.
Let $M' = \struct {A', d'}$ be a metric space.
Let $M$ be homeomorphic to $M'$.
Then it is not necessarily the case that $M'$ is totally bounded.
== Proof ==
== Sources ==
* : Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces
Category:Homeomorphisms (Metric Spaces)
Category:Totally Bounded Metric Spaces
Total Charge carried By Electron in Hydrogen Atom

== Theorem ==
Consider an atom of hydrogen $\mathrm H$.
Then:
:$\ds \int_V \map {\rho_{\mathrm {el} } } {\mathbf r} \rd \tau = -\E$
where:
:$\d \tau$ is an infinitesimal volume element
:$\mathbf r$ is the position vector of $\d \tau$
:$V$ is a volume large enough to completely enclose $\mathrm H$
:$\map {\rho_{\mathrm {el} } } {\mathbf r}$ is the electric charge density caused by the charge on the electron in the electron cloud at $\mathbf r$
:$\E$ is the elementary charge.
=== Cartesian Form ===
== Proof ==
The total electric charge on $\mathrm H$ carried by the electron is equal to the total charge on the electron.
By definition of the charge on the electron, this total is $-\E$.
The result follows.
== Sources ==
* : Chapter $1$: Force and energy in electrostatics: $1.3$ Electric Fields in Matter: $1.3.1$ The atomic charge density
Category:Total Charge carried By Electron in Hydrogen Atom
Category:Hydrogen
Category:Electrons
Total Charge carried By Electron in Hydrogen Atom/Cartesian Form

== Theorem ==
Consider an atom of hydrogen $\mathrm H$.
The total electric charge on $\mathrm H$ carried by the electron can be expressed in Cartesian coordinates as:
== Proof ==
Follows directly.
== Sources ==
* : Chapter $1$: Force and energy in electrostatics: $1.3$ Electric Fields in Matter: $1.3.1$ The atomic charge density
Category:Total Charge carried By Electron in Hydrogen Atom
Total Curvature Theorem

== Theorem ==
Let $\closedint a b \subseteq \R$ be a closed real interval.
Let $\gamma: \closedint a b \to \R^2$ be a smooth unit-speed curve on a plane.
Let $N$ be the inward pointing normal.
Let $\map {\kappa_N} t$ be the curvature of $\gamma$ at $\map \gamma t$.
Suppose $\map {\gamma'} a = \map {\gamma'} b$.
Then:
:$\ds \int_a^b \map {\kappa_N} t \rd t = 2 \pi$
== Proof ==
== Sources ==
* : $\S 1$: What Is Curvature? The Euclidean Plane
Category:Curves
Category:Curvature
Total Electric Flux through Surface

== Physical Law ==
Let $\mathbf E$ be an electric field which acts on a region of space $R$.
Let $S$ be a surface embedded in $R$.
The total electric flux through $S$ is the surface integral over $S$ of the dot product of $\mathbf E$ with the vector area of $S$:
:$F = \ds \int_S \mathbf E \cdot \rd \mathbf S$
where $\d \mathbf S$ is an infinitesimal area element of $S$.
== Proof ==
Let $S$ be divided up into area elements of vector areas $\delta \mathbf S$, where the sign of each area element is defined in the same sense.
The electric flux through $\delta \mathbf S$ is defined as:
:$\mathbf E \cdot \delta \mathbf S$
Summing all of these, we get:
:$F = \ds \sum_{\text {all surfaces $\delta \mathbf S$} } \mathbf E \cdot \delta \mathbf S$
As $\ds \lim_{\text {area of $\delta \mathbf S$} } \to 0$, this becomes the surface integral over $S$:
:$F = \ds \int_S \mathbf E \cdot \rd \mathbf S$
== Sources ==
* : Chapter $1$: Force and energy in electrostatics: $1.4$ Gauss's Law: $1.4.1$ The flux of a vector field
Category:Electric Flux
Total Energy of Body in Motion

== Theorem ==
Let $B$ be a body in motion in space.
The total energy $E$ of $B$ is given by:
:$E^2 = p^2 c^2 + {m_0}^2 c^4$
where:
:$p$ is the magnitude of the linear momentum of $B$
:$m_0$ is the rest mass of $B$
:$c$ is the speed of light.
== Proof ==
== Sources ==
* 
* 
Category:Energy
Category:Relativistic Mechanics
Total Expectation Theorem

== Theorem ==
Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a discrete random variable on $\EE$.
Let $\set {B_1 \mid B_2 \mid \cdots}$ be a partition of $\Omega$ such that $\map \Pr {B_i} > 0$ for each $i$.
Then:
:$\ds \expect X = \sum_i \expect {X \mid B_i} \, \map \Pr {B_i}$
whenever this sum converges absolutely.
In the above:
:$\expect X$ denotes the expectation of $X$
:$\expect {X \mid B_i}$ denotes the conditional expectation of $X$ given $B_i$.
== Proof ==
== Also known as ==
== Sources ==
* : $\S 2.5$: Conditional expectation and the partition theorem: Theorem $2 \ \text C$
Category:Total Expectation Theorem
Category:Expectation
Category:Named Theorems
Total Expectation Theorem/Also known as

== Total Expectation Theorem: Also known as ==
Some sources refer to the Total Expectation Theorem as the Partition Theorem, which causes ambiguity, as that name is used for other things as well.
Some sources give this as the Law of Total Expectation.
== Sources ==
* : $\S 2.5$: Conditional expectation and the partition theorem: Theorem $2 \ \text C$
Category:Total Expectation Theorem