nyuuzyou/wikis huggingface dataset:
Nested Sequences in Complete Metric Space not Tending to Zero may be Disjoint

== Theorem ==
Let $M = \struct {A, d}$ be a complete metric space.
Let $\family {S_k}_{k \mathop \in \N}$ be a nested sequence of closed balls in $M$.
Let the radii of $\family {S_k}_{k \mathop \in \N}$ be convergent in $M$, but not to zero.
Then it is not necessarily the case that their intersection $\ds \bigcap S_k$ is non-empty.
== Proof ==
Let $M = \struct {A, d}$ be Sierpinski's metric space:
:$A = \set {x_i: i = 1, 2, 3, \ldots}$
:$\map d {x_i, x_j} = 1 + \dfrac 1 {i + j}$
Let $S_k = \set {y \in A: \map d {y, x_k} \le 1 + \dfrac 1 {2 n} }$.
From Nested Sequence of Closed Balls in Sierpinski's Metric Space with Empty Intersection:
:$\ds \bigcap S_k = \O$
== Sources ==
* : Part $\text I$: Basic Definitions: Section $5$: Metric Spaces: Complete Metric Spaces
Category:Complete Metric Spaces
Category:Nested Sequences
Nested Sphere Theorem

== Theorem ==
Let $M = \struct {A, d}$ be a complete metric space.
Let $\sequence {S_n}$ be a sequence of closed balls in $M$ defined by:
:$S_n = \map {B^-_{\rho_n} } {x_n}$
where $\rho_n \to 0$ as $n \to \infty$ and:
:$S_1 \supseteq S_2 \supseteq \cdots \supseteq S_n \supseteq \cdots$
Then there exists $x \in A$ such that:
:$\ds \bigcap_{n \mathop = 1}^\infty S_n = \set x$
== Proof ==
Let $S_n = \map {B^-_{\rho_n} } {x_n}$ be the closed ball of radius $\rho_n$ about the point $x_n$.
That is, let $S_n = \set {x \in A: \map d {x_n, x} \le \rho_n}$.
Then the sequence $\sequence {x_n}$ forms a Cauchy sequence:
:$\map d {x_n, x_{n + p} } < \rho_n$
for any $p \ge 0$ since $S_{n + p} \subseteq S_n$.
However, $\rho_n \to 0$ as $n \to \infty$ and therefore $\map d {x_n, x_{n + p} } \to 0$ as $n \to \infty$ for any $p \ge 0$.
Since the space $M$ is complete, there exists $x \in X$ such that $x_n \to x$ as $n \to \infty$.
Then since the subsequence $\sequence {x_k}_{k \mathop = n}^\infty$ is contained entirely in $S_n$ and converges to $x$, $x$ is a member of the closure of $S_n$ by Closure of Subset of Metric Space by Convergent Sequence.
Since $S_n$ is closed, by Closed Set Equals its Closure, we have:
:$\forall n \in \N: x \in S_n$
Hence:
:$\ds x \in \bigcap_{n \mathop = 1}^\infty S_n$
Suppose now that $y \ne x$.
Then it follows that $\map d {x, y} > 0$, and hence that for some $n$:
:$\map d {x, y} > 2 \rho_n$
Since $x, x_n \in S_n$, it then follows that:
:$\map d {x, x_n} \le \rho_n$
Now:
From the above, it follows that:
:$\map d {x_n, y} > \rho_n$
so that $y \notin S_n$, and consequently:
:$\ds y \notin \bigcap_{i \mathop = 1}^\infty S_n$
Hence:
:$\ds \bigcap_{n \mathop = 1}^\infty S_n = \set x$
Category:Complete Metric Spaces
Category:Named Theorems
Nesthood has Finite Character

== Theorem ==
Let $P$ be the property of sets defined as:
:$\forall x: \map P x$ denotes that $x$ is a nest.
Then $P$ is of finite character.
That is:
:$x$ is a nest
:
:every finite subset of $x$ is a nest.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text {II}$ -- Maximal principles: $\S 5$ Maximal principles
Category:Nests
Category:Finite Character
Category:Nesthood has Finite Character
Nesthood has Finite Character/Proof 2

== Theorem ==
== Proof ==
By definition, a nest is a class on which $\subseteq$ is a total ordering.
=== Sufficient Condition ===
Let $x$ be a nest.
Let $y \subseteq x$.
From Restriction of Total Ordering is Total Ordering it follows that $y$ is also a nest.
This holds in particular if $y$ is a finite set.
Hence the result.
=== Necessary Condition ===
Suppose that every finite subset of $x$ is a nest.
We have that Subset Relation is Ordering.
It remains to show that $\subseteq$ is total.
Let $y, y' \in x$ be arbitrary.
Since $\set {y, y'}$ is a finite subset of $x$, it is a nest.
Therefore, either $y \subseteq y'$ or $y' \subseteq y$.
Since $y, y'$ were arbitrary, it follows that $x$ is also a nest.
Category:Nesthood has Finite Character
Net Charge within Electrically Neutral Body of Matter

== Definition ==
Let $B$ be a body of matter.
Let $B$ be neutral.
Let $\map {\rho_{\text {atomic} } } {\mathbf r}$ denote the atomic charge density at a point within $B$ whose position vector is $\mathbf r$.
Then:
:$\ds \int_B \map {\rho_{\text {atomic} } } {\mathbf r} \rd V = 0$
where $\d V$ denotes an infinitesimal volume element containing the point whose position vector is $\mathbf r$.
== Proof ==
By definition, $\ds \int_B \map {\rho_{\text {atomic} } } {\mathbf r} \rd V$ denotes the total electric charge on $B$.
Hence the result.
== Sources ==
* : Chapter $1$: Force and energy in electrostatics: $1.3$ Electric Fields in Matter: $1.3.1$ The atomic charge density
Category:Electric Charge Density
Category:Atoms
Net Convergence Equivalent to Absolute Convergence

== Theorem ==
Let $V$ be a Banach space.
Let $\sequence {v_n}_{n \mathop \in \N}$ be a sequence of elements in $V$.
Then the following two statements are equivalent:
:$(1): \quad \ds \sum_{n \mathop = 1}^\infty \norm {v_n}$ converges (absolute convergence)
:$(2): \quad \ds \sum \set {v_n: n \in \N}$ converges (generalised or net convergence)
== Proof ==
== Sources ==
* $\text {I}.4$ Exercise $10 - 12$
Category:Banach Spaces
Category:Nets (Set Theory)
Network Analysis/Examples/Arbitrary Example 1

== Example of Network Analysis ==
The following is an example of an exercise in network analysis:
:
Each vertex of the above network represents a town.
Each edge represents a permissible route to travel between each town.
The number assigned to each edge represents the cost to make the journey using that route.
The exercise is to determine the cheapest route between each pair of towns.
Clearly, for example, it is cheaper to travel from $A$ to $D$ via $C$ and $F$ rather than go there directly.
== Sources ==
* 
* 
Category:Examples of Network Analysis
Network with Positive Integer Mapping is Multigraph

== Theorem ==
Let $N = \struct {V, E, w}$ be a network whose weights are all strictly positive integers.
Then $N$ can be represented as a multigraph.
Conversely, any multigraph can be expressed as a network whose weights are all strictly positive integers.
== Proof ==
=== Network as Multigraph ===
Let $N = \struct {V, E, w}$ be a network, either directed or undirected.
, suppose $N$ is undirected, and in the following argument allow the term edge to include arcs.
Let $e \in E$ be an edge of the underlying graph $\struct {V, E}$ of $N$.
Let $e = u v$, where $u, v \in V$ are vertices of the underlying graph $\struct {V, E}$ of $N$.
Let $n = \map w e$ be the weight of $e$.
By hypothesis, $n$ is a strictly positive integer.
Let $n$ edges be identified between $u$ and $v$.
Repeat this process for all edges of $N$.
The resulting object is a multigraph.
=== Multigraph as Network ===
Let $G = \struct {V, E}$ be a multigraph, where each element of $E$ may occur multiple times.
Let $e = u v$ where $e \in E$ and $u, v \in V$.
Let $\map m e$ be the multiplicity of $e$.
Then $m$ can be identified with a weight function from $w: E \to \Z_{>0}$.
Thus the network $N = \struct {V, E, w}$ can be created.
== Examples ==
== Sources ==
* : Chapter $1$: Mathematical Models: $\S 1.6$: Networks as Mathematical Models
Category:Network Theory
Category:Graph Theory
Category:Network with Positive Integer Mapping is Multigraph
Network with Positive Integer Mapping is Multigraph/Examples/Arbitrary Order 3 Network

== Example of Use of Network with Positive Integer Mapping is Multigraph ==
Let $G = \struct {V, E}$ be the undirected graph defined as:
:$V = \set {v_1, v_2, v_3}$
:$E = \set {v_1 v_2, v_1 v_3, v_2 v_3}$
Let $w: E \to \R$ be the weight function defined as:
:$w = \set {\tuple {v_1 v_2, 2}, \tuple {v_1 v_3, 1}, \tuple {v_2 v_3, 3} }$
Then the undirected network $\struct {G, w}$ can be represented diagrammatically in either of the forms shown below:
:
== Sources ==
* : Chapter $1$: Mathematical Models: $\S 1.6$: Networks as Mathematical Models
Category:Network with Positive Integer Mapping is Multigraph
Neumann Eigenvalue is Nonnegative

== Theorem ==
Let $\struct {M, g}$ be a compact connected Riemannian manifold with non-empty boundary $\partial M$.
Let $\lambda$ be a Neumann eigenvalue of $M$.
Then $\lambda$ is non-negative.
That is, $0$ is a Neumann eigenvalue, and all the other Neumann eigenvalues are strictly positive.
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Problems
Category:Riemannian Geometry
Neumann Series Theorem

== Theorem ==
Let $X$ be a Banach space.
Let $\map {CL} X$ be the continous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Let $A \in \map {CL} X$ be such that $\norm A < 1$.
Let $\circ$ be the composition of mappings.
Let $I$ be the identity mapping.
Then:
:$I - A$ is invertible in $\map {CL} X$
:$\ds \paren {I - A}^{-1} = \sum_{n \mathop = 0}^\infty A^n$
:$\norm {\paren{I - A}^{-1} } \le \dfrac 1 {1 - \norm A}$
=== Corollary 1 ===
=== Corollary 2 ===
== Proof ==
Let $\ds S_k := \sum_{n \mathop = 0}^k A^n$.
=== $\ds \sum_{n \mathop = 0}^\infty A^n$ converges absolutely ===
By Supremum Operator Norm on Continuous Linear Transformation Space is Submultiplicative:
:$\forall n \in \N : \norm {A^n} \le \norm A^n$
By assumption, $\norm A < 1$.
By Sum of Infinite Geometric Sequence, $\ds \sum_{n \mathop = 0}^\infty \norm A^n$ converges.
By series comparison, $\ds \sum_{n \mathop = 0}^\infty \norm {A^n}$ converges too.
By definition, $\ds \sum_{n \mathop = 0}^\infty A^n$ is absolutely convergent.
=== $\ds \sum_{n \mathop = 0}^\infty A^n$ converges ===
By assumption, $X$ is Banach.
By Necessary and Sufficient Conditions for Continuous Linear Transformation Space to be Banach Space, $\map {CL} X$ is Banach too.
Let $\ds S := \sum_{n \mathop = 0}^\infty A^n$.
We have that Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach.
Then $S$ converges in $\map {CL} X$.
In other words:
:$\ds \lim_{k \mathop \to \infty} S_k = S$
=== Inverse of $\paren {I - A}$ is $S$ ===
We have that:
Furthermore:
Hence:
That is:
:$S A = A S = S - I$
It follows that:
Therefore, $I - A$ is invertible in $\map {CL} X$.
Furthermore:
:$\ds \paren {I - A}^{-1} = S = \sum_{k \mathop = 0}^\infty A^k$
Moreover:
== Also see ==
* Definition:Neumann Series
== Sources ==
* : Chapter $\S 2.4$: Composition of continuous linear transformations
Category:Neumann Series
Category:Continuous Linear Transformations
Category:Banach Spaces
Neumann Series Theorem/Corollary 1

== Theorem ==
Let $X$ be a Banach space.
Let $\map {CL} X$ be the continous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Let $A \in \map {CL} X$ be such that $\norm A < 1$.
Let $I$ be the identity mapping.
The mapping $I - A : X \to X$ is bijective.
== Proof ==
== Sources ==
* : Chapter $\S 2.4$: Composition of continuous linear transformations
Category:Operators
Category:Banach Spaces
Category:Bijections
Neumann Series Theorem/Corollary 2

== Theorem ==
Let $X$ be a Banach space.
Let $\map {CL} X$ be the continous linear transformation space.
Let $\norm {\, \cdot \,}$ be the supremum operator norm.
Let $A \in \map {CL} X$ be such that $\norm A < 1$.
Let $I$ be the identity mapping.
The mapping $\paren {I - A}^{-1} : X \to X$ is continuous.
== Proof ==
== Sources ==
* : Chapter $\S 2.4$: Composition of continuous linear transformations
Category:Operators
Category:Banach Spaces
Category:Bijections
Newton's Formula for Pi

== Theorem ==
$\pi$ (pi) can be approximated using the formula:
:$\pi = \dfrac {3 \sqrt 3} 4 + 24 \paren {\dfrac 2 {3 \times 2^3} - \dfrac 1 {5 \times 2^5} - \dfrac 1 {28 \times 2^7} - \dfrac 1 {72 \times 2^9} - \dfrac 5 {704 \times 2^{11} } - \dfrac 7 {1664 \times 2^{13} } - \cdots}$
== Proof ==
Let $\AA$ denote the area of the shaded region in the following diagram:
:
Consider the semicircle embedded in the cartesian plane:
:whose radius is $\dfrac 1 2$
and
:whose center is the point $\tuple {\dfrac 1 2, 0}$. 
We have:
We calculate $\AA$ in $2$ different ways.
First we calculate it as the definite integral between $x = 0$ and $x = \dfrac 1 4$:
Then we calculate $\AA$ using the techniques of plane geometry.
From the construction, we have that:
:$AC = CD$
:$AB = BC$
:$BD$ is common
so by Triangle Side-Side-Side Congruence:
:$\triangle ABD = \triangle CBD$
Thus:
:$AD = AC = CD$
and so $\triangle ACD$ is equilateral.
Thus we have that $\triangle BCD$ has angles $30 \degrees$, $60 \degrees$ and $90 \degrees$.
Hence by Pythagoras's Theorem:
Then we observe that $\AA$ is:
:the area $\AA_S$ of the sector of the semicircle whose central angle is $\dfrac \pi 3$
minus:
:the area $\AA_T$ of the right triangle $\triangle BCD$.
$\AA_S$ is $\dfrac 1 6$ of the area of the circle whose radius is $\dfrac 1 2$.
Thus from Area of Circle:
:$\AA_S = \dfrac 1 6 \paren {\pi \paren {\dfrac 1 2}^2}$
$\AA_T$ is given from Area of Triangle in Terms of Side and Altitude as:
:$\AA_T = \dfrac 1 2 \paren {\dfrac 1 4} \paren {\dfrac {\sqrt 3} 4}$
Hence:
Finally, setting $(1)$ equal to $(2)$:
== Historical Note ==
== Sources ==
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
* : Chapter $7$: A Gem from Isaac Newton (Late 1660s): Great Theorem: Newton's Approximation of $\pi$
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
Category:Formulas for Pi
Newton's Formula for Pi/Historical Note

== Historical Note on Newton's Formula for Pi ==
 arrived at his formula for $\pi$ after having returned to his home in Grantham in $1666$ to escape the epidemic of bubonic plague.
He used it to find $\pi$ to $16$ places by using only $22$ terms of his formula.
== Sources ==
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
Category:Historical Notes
Newton's Identities

== Theorem ==
Let $\PP$ be the polynomial equation:
:$\PP: \quad a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x^1 + a_0 = 0$
Let $s_k$ be the sum of the $k$th powers of the roots of $\PP$.
Then Newton's Identities are:
where $a_i$ is taken to be $0$ if $i < 0$.
== Proof ==
== Also see ==
* Newton-Girard Identities, also known as Newton's Identities
== Historical Note ==
== Sources ==
* 
Category:Polynomial Theory
Newton's Law of Cooling/Historical Note

== Historical Note on Newton's Law of Cooling ==
 applied this law to make an estimate of the temperature of a red-hot iron ball.
Although this approximation was somewhat crude, it was better than anything else up till then.
== Sources ==
* : $1$: The Nature of Differential Equations: $\S 4$: Growth, Decay and Chemical Reactions: Problem $5$: Footnote $^1$
Category:Historical Notes
Newton's Law of Restitution

== Physical Law ==
Let two solid bodies $B_1$ and $B_2$ be in collision.
Let the components of the relative velocities of $B_1$ and $B_2$ in the direction of their common normal be respectively:
:$\mathbf u_1$ and $\mathbf u_2$ before the collision
:$\mathbf v_1$ and $\mathbf v_2$ after the collision.
Then:
:$\mathbf v_2 - \mathbf v_1 = -e \paren {\mathbf u_2 - \mathbf u_1}$
where $e$ the coefficient of restitution.
== Historical Note ==
== Sources ==
* 
* 
Category:Newton's Laws
Category:Mechanics
Newton's Law of Universal Gravitation

== Physical Law ==
Let $a$ and $b$ be particles with mass $m_a$ and $m_b$ respectively.
Then $a$ and $b$ exert a force upon each other whose magnitude and direction are given by Newton's Law of Universal Gravitation:
:$\mathbf F_{a b} \propto \dfrac {m_a m_b {\mathbf r_{b a} } } {r^3}$
where:
:$\mathbf F_{a b}$ is the force exerted on $b$ by the gravitational force on $a$
:$\mathbf r_{b a}$ is the displacement vector from $b$ to $a$
:$r$ is the distance between $a$ and $b$.
Thus it is seen that the direction of $\mathbf F_{a b}$ is specifically towards $a$.
By exchanging $a$ and $b$ in the above, it is seen that $b$ exerts the same force on $a$ as $a$ does on $b$, but in the opposite direction, that is, towards $b$.
=== Universal Gravitational Constant ===
Thus the equation becomes:
:$\mathbf F_{a b} = \dfrac {G m_a m_b \mathbf r_{b a} } {r^3}$
== Also presented as ==
== Also known as ==
== Also see ==
* Definition:Gravitational Field
== Historical Note ==
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : $\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: The Gravitational Constant
* : $\S 3.21$: Newton's Law of Gravitation: $(11)$
* 
* : Chapter $1$: Force and energy in electrostatics
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation
* 
* 
* 
* 
* 
* 
* : Chapter $8$: The System of the World: Newton
* 
Category:Newton's Law of Universal Gravitation
Category:Gravitation
Category:Newton's Laws
Newton's Law of Universal Gravitation/Also known as

== Newton's Law of Universal Gravitation: Also known as ==
Newton's Law of Universal Gravitation is also known as:
*Newton's Law of Gravitation
* the Inverse Square Law of Gravitation
*Newton's Inverse Square Law of Universal Gravitation
and so on.
Some sources refer to it as Newton's Law but there are a number of these.
== Sources ==
* : $\S 3.21$: Newton's Law of Gravitation: $(11)$
* 
* : Chapter $1$: Force and energy in electrostatics
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation
* 
* 
* 
* 
* 
Category:Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation/Also presented as

== Newton's Law of Universal Gravitation: Also presented as ==
Newton's Law of Universal Gravitation can also be presented in the form:
:$\mathbf F_{a b} \propto \dfrac {m_a m_b \hat {\mathbf r}_{b a} } {r^2}$
where $\hat {\mathbf r}_{a b}$ is the unit vector in the direction from $b$ to $a$.
Some presentations do not consider the vector aspects, merely expressing the relationship in terms of scalars:
:$F = \dfrac {k m_a m_b} {r^2}$
where the constant is variously specified.
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : $\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: The Gravitational Constant
* : $\S 3.21$: Newton's Law of Gravitation: $(11)$
* 
* : Chapter $\text {B}.25$: Kepler's Laws and Newton's Law of Gravitation
Category:Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation/Historical Note

== Historical Note on Newton's Law of Universal Gravitation ==
The popular tale has it that had the idea while lying in the garden at his home in Woolsthorpe between the years of $1665$ and $1667$ and watching an apple fall from a tree.
This tale has the feel of an off-the-cuff comment that may have made during the course of an interview with a journalist of the time, or whatever the equivalent may have been.
The children's version of the tale has it that the apple hit him on the head.
However, the inverse square law was in fact conjectured by in $1684$, but he was unable to do anything to prove his conjecture. He discussed this with and , who claimed he had a proof of it. However, disbelieved him.
Some months later, had the chance to ask what law of attraction would cause the planets to move in an elliptical orbit. answered immediately that it would be an inverse square law, and claimed to have already calculated it.
Some sources state that the question was posed the other way round: that asked how the planets would behave under a central force obeying the inverse square law.
The mathematical work that he had performed to prove had supposedly been deduced from Kepler's Laws of Planetary Motion.
Having been thus spurred on by , went ahead to write and publish his .
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : $\text {I}$: Motion, Sound and Heat: Chapter $4$: Gravitation: Combination of Forces
* : $\S 3$: Appendix $\text B$: Newton
* : Chapter $1$: Force and energy in electrostatics
* : Chapter $\text {A}.18$: Newton ()
* : Chapter $8$: The System of the World: Newton
Category:Historical Notes
Newton's Laws of Motion

== Physical Laws ==
Newton's Laws of Motion are a set of three physical laws that form the basis for classical mechanics.
=== First Law ===
=== Second Law ===
=== Third Law ===
== Also known as ==
== Historical Note ==
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : $\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Inertia
* 
* 
* 
Category:Newton's Laws of Motion
Category:Newton's Laws
Category:Physics
Category:Classical Mechanics
Category:Force
Newton's Laws of Motion/First Law

== Physical Law ==
Newton's First Law of Motion is one of three physical laws that forms the basis for classical mechanics.
=== Statement of Law ===
:Every body remains in a state of constant velocity unless it is acted upon by an external unbalanced force. 
This of course includes it being stationary, that is, with a constant velocity of zero.
This tendency is sometimes called inertia.
As himself put it:
:A body remains at rest or, if already in motion, remains in uniform motion with constant speed in a straight line, unless it is acted on by an unbalanced external force.
== Also known as ==
== Also see ==
* Newton's Second Law of Motion
* Newton's Third Law of Motion
It is worth pointing out that Newton's First Law of Motion is in fact a special case of Newton's Second Law of Motion, putting $\mathbf F = 0$.
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : Chapter $\text {I}$: Introduction: $(1)$
* : $\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Inertia
* : Chapter $1$ Special Principle of Relativity. Lorentz Transformations: $1$. Newton's laws of motion
* 
* 
* 
Category:Newton's Laws of Motion
Newton's Laws of Motion/Second Law

== Physical Law ==
Newton's Second Law of Motion is one of three physical laws that forms the basis for classical mechanics.
=== Statement of Law ===
The total force applied on a body is equal to the derivative with respect to time of the linear momentum of the body:
== Also presented as ==
== Also known as ==
== Also see ==
* At velocities near the speed of light, see Einstein's Law of Motion.
* Newton's First Law of Motion
* Newton's Third Law of Motion
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : Chapter $\text {I}$: Introduction: $(2)$
* : Chapter $1$ Ordinary Differential Equations: $1.1$ Introduction
* : Chapter $\text {B}.7$: A Simple Approach to $E = M c^2$
* : Chapter $\text {B}.8$: Rocket Propulsion in Outer Space
* 
* 
* 
* 
* 
* 
* 
Category:Newton's Laws of Motion
Newton's Laws of Motion/Third Law

== Physical Law ==
Newton's third law of motion is one of three physical laws that forms the basis for classical mechanics.
=== Statement of Law ===
:To every force there is always an equal and opposite force.
:That is, the forces of two bodies on each other are always equal and are directed in opposite directions.
As himself put it:
:Whenever one body exerts a force on a second body, the second body exerts a force on the first body. These forces are equal in magnitude and opposite in direction.
== Also known as ==
== Also see ==
* Newton's First Law of Motion
* Newton's Second Law of Motion
== Sources ==
* : Chapter $\text{VI}$: On the Seashore
* : Chapter $\text {I}$: Introduction: $(3)$
* : $\text {I}$: Motion, Sound and Heat: Chapter $3$: The Laws of Motion: Action and Reaction
* 
* 
* 
* 
* 
Category:Newton's Laws of Motion