nyuuzyou/wikis huggingface dataset:
Cumulative Distribution Function as Integral of Probability Density Function

== Theorem ==
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an absolutely continuous random variable. 
Let $f_X$ be a probability density function for $X$. 
Let $F_X$ be the cumulative distribution function for $X$. 
Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$. 
Let $\lambda$ be the Lebesgue measure on $\struct {\R, \map \BB \R}$. 
Then: 
:$\ds \map {F_X} x = \int_{-\infty}^x f_X \rd \lambda$
for each $x \in \R$, where $\ds \int_{-\infty}^x f_X \rd \lambda$ denotes the Lebesgue integral of $f$ over $\hointl {-\infty} x$.
== Proof ==
Let $P_X$ be the probability distribution of $X$. 
Since $f_X$ is a probability density function for $X$, $f_X$ is a Radon-Nikodym derivative of $P_X$ with respect to $\lambda$. 
Then, we have: 
== Sources ==
* 
* 
Category:Cumulative Distribution Functions
Category:Probability Density Functions
Cumulative Distribution Function for Discrete Distribution

== Theorem ==
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.
Then the cumulative distribution function of $X$ is given by:
:$\map F x = \ds \sum_{x_i \mathop \le x} \map \Pr {X = x_i}$
== Proof ==
== Sources ==
* 
* 
Category:Cumulative Distribution Functions
Category:Discrete Probability Distributions
Cumulative Distribution Function is Increasing

== Theorem ==
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $F_X$ be the cumulative distribution function.
Then:
:$F_X$ is an increasing function.
== Proof ==
Let $x, y \in \R$ have $x \le y$. 
Note that if $\omega \in \Omega$ is such that: 
:$\map X \omega \le x$ 
then: 
:$\map X \omega \le y$
so:
:$\set {\omega \in \Omega : \map X \omega \le x} \subseteq \set {\omega \in \Omega : \map X \omega \le y}$
From Measure is Monotone, we then have: 
:$\map \Pr {X \le x} \le \map \Pr {X \le y}$
That is, from the definition of cumulative distribution function, we have: 
:$\map {F_X} x \le \map {F_X} y$ 
whenever $x \le y$. 
So:
:$F_X$ is an increasing function.
== Sources ==
* 
* 
Category:Cumulative Distribution Functions
Cumulative Distribution Function is Right-Continuous

== Theorem ==
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. 
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $F_X$ be the cumulative distribution function of $X$. 
Then:
:$F_X$ is right-continuous. 
== Proof ==
Let $x \in \R$. 
We show that $F_X$ is right-continuous at $x$. 
We use Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals, and will show that: 
:for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$, with $x_n > x$ for each $n$, that converge to $x$ we have:
::$\map {F_X} {x_n} \to \map {F_X} x$
Let $\sequence {x_n}_{n \mathop \in \N}$ be a monotone sequences with $x_n > x$ for each $n$ that converges to $x$.
Then $\sequence {x_n}_{n \mathop \in \N}$ is a decreasing sequence. 
So: 
:$\sequence {\hointl {-\infty} {x_n} }_{n \mathop \in \N}$ is a decreasing sequence of sets.
From Limit of Decreasing Sequence of Unbounded Below Closed Intervals: 
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \hointl {-\infty} x$
Let $P_X$ be the probability distribution of $X$. 
Note that $P_X$ is a finite measure. 
So, from Measure of Limit of Decreasing Sequence of Measurable Sets, we therefore have: 
:$\ds \map {P_X} {\hointl {-\infty} x} = \lim_{n \mathop \to \infty} \map {P_X} {\hointl {-\infty} {x_n} }$
So we obtain: 
So, since $\sequence {x_n}_{n \mathop \in \N}$ was arbitrary:
:for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$, with $x_n > x$ for each $n$, that converge to $x$ we have:
::$\map {F_X} {x_n} \to \map {F_X} x$
So, from Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals:
:$F_X$ is right-continuous at $x$.
Category:Right-Continuous Functions
Category:Cumulative Distribution Functions
Category:Cumulative Distribution Function is Right-Continuous
Cumulative Distribution Function of Logistic Distribution

== Theorem ==
Let $X$ be a continuous random variable with the logistic distribution.
Then the cumulative distribution function of $X$ is:
:$\map {F_X} x = \dfrac 1 {1 + \map \exp {- \dfrac {x - \mu} s} }$
== Proof ==
The derivative of $F_X$ is:
By the Fundamental Theorem of Calculus:
:$\ds \int_a^b \map {f_X} x \rd x = \bigintlimits {\map {F_X} x} {x \mathop = a} {x \mathop = b}$
Therefore:
Hence the result.
Category:Logistic Distribution
Cumulative Distribution of Random Sample

== Theorem ==
Let $S = \tuple {x_1, x_2, \ldots, x_n}$ be a random sample from a probability distribution either discrete or continuous, ordered by value.
Then the cumulative distribution of $S$ is a step function with step $\dfrac 1 n$ at each $x_i$.
== Proof ==
== Sources ==
* 
* 
Category:Random Samples (Probability Theory)
Cumulative Rounding Error/Examples/Illustration of Round to Even

== Example of Cumulative Rounding Error ==
Let $S$ be the following set of numbers reported to $2$ decimal places:
:$S = \set {4.35, 8.65, 2.95, 12.45, 6.65, 7.55, 9.75}$
The sum $\sum S$ of the elements of $S$ is:
:$\sum S = 4.35 + 8.65 + 2.95 + 12.45 + 6.65 + 7.55 + 9.75 = 52.35$
We desire to round the elements of $S$ to $1$ decimal place before adding them.
We need to decide which strategy to use for the treatment of the half:
:rounding up, so, for example, $4.35 \to 4.4$ and $12.45 \to 12.5$
:rounding down, so, for example, $4.35 \to 4.3$ and $12.45 \to 12.4$
:rounding to even, so, for example, $4.35 \to 4.4$ and $12.45 \to 12.4$
First, we use the strategy of rounding up.
Let $S_u$ be the set consisting of the elements of $S$ rounded up:
:$S_u = \set {4.4, 8.7, 3.0, 12.5, 6.7, 7.6, 9.8}$
:$\sum {S_u} = 4.4 + 8.7 + 3.0 + 12.5 + 6.7 + 7.6 + 9.8 = 52.7$
Next, we use the strategy of rounding down.
Let $S_d$ be the set consisting of the elements of $S$ rounded down:
:$S_d = \set {4.3, 8.6, 2.9, 12.4, 6.6, 7.5, 9.7}$
:$\sum {S_d} = 4.3 + 8.6 + 2.9 + 12.4 + 6.6 + 7.5 + 9.7 = 52.0$
Next, we use the strategy of rounding to even.
Let $S_e$ be the set consisting of the elements of $S$ rounded to even:
:$S_e = \set {4.4, 8.6, 3.0, 12.4, 6.6, 7.6, 9.8}$
:$\sum {S_e} = 4.4 + 8.6 + 3.0 + 12.4 + 6.6 + 7.6 + 9.8 = 52.4$
As can be seen, rounding to even gets us closest to the true value.
== Sources ==
* : Chapter $1$: Variables and Graphs: Solved Problems: Rounding of Data: $1.4$
Category:Examples of Rounding
Curl Operator/Examples/Magnetic Field of Conductor

== Example of Curl Operator ==
Consider a conductor of electricity $C$.
Let $C$ be carrying a steady current $I$.
Let $P$ be an arbitrary point in the magnetic field $\mathbf H$ induced by $I$.
Let a small plane surface be placed at $P$, turned into a position so that the contour integral of the magnetic force taken around its boundary is the greatest possible.
This value per unit area is the curl of $\mathbf H$.
This is the magneto-motive force per unit area at $P$.
If $P$ is within $C$ at the point where current density is $\mathbf J$, this will be the total current passing normally through the closed curve when the contour integral is greatest.
We have from the Ampère-Maxwell Law that:
:$\curl \mathbf H = \mathbf J$
For a point in the magnetic field outside the conductor there is no current density and so:
:$\curl \mathbf H = 0$
== Sources ==
* : Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $5$. Simple Examples of Curl
Category:Curl Operator
Category:Electromagnetism
Curl Operator/Examples/Motion of Fluid

== Example of Curl Operator ==
Consider an infinitesimal volume of fluid $F$.
It may have $3$ kinds of motion:
:$(1): \quad$ Moving with a linear velocity as a whole
:$(2): \quad$ Changing its shape
:$(3): \quad$ In rotary motion.
At any instant, $F$ may be regarded as a rigid body.
Hence from Curl of Rotation of Rigid Body, the curl of the velocity of $F$ is twice its angular velocity where its axis of rotation at that instant is the same as that of the curl.
:
Consider the diagram above.
On the left, the element $E_1$ has itself rotated in moving to ${E_1}'$.
If every element of the body of fluid has rotated the same amount, $\curl \mathbf V$ would be twice the angular velocity about $O$.
On the right, on the other hand, the element $E_2$ has not actually rotated in moving to ${E_2}'$.
Hence there is no $\curl \mathbf V$ and its angular velocity is zero.
== Sources ==
* : Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $5$. Simple Examples of Curl
Category:Curl Operator
Category:Fluid Mechanics
Curl Operator/Examples/Rotation of Rigid Body

== Example of Curl Operator ==
Consider a rigid body $B$ in rotary motion with angular velocity $\omega$ about an axis $OA$, where $O$ is some fixed point inside $B$.
Let $P$ be an arbitrary point inside $B$.
Let $B$ also be subject to a linear velocity $\mathbf v_0$ in an arbitrary direction.
Let the instantaneous velocity of $P$ be $\mathbf V$
Then:
:$\bsomega = \dfrac 1 2 \curl \mathbf V$
where $\bsomega$ is the angular velocity (axial) vector along the axis $OA$ in the sense according to the right-hand rule.
== Proof ==
$P$ moves in a circular path $C$ around $OA$ with an instantaneous velocity $v$, tangent to $C$ where:
:$r$ is the distance of $P$ from $O$
:$\theta$ is the angle between $OR$ and $OP$.
:
As noted, the angular velocity is a vector (axial) $\bsomega$ along the axis $OA$ in the sense according to the right-hand rule.
Let the position vector of $P$ be given by $\mathbf r$.
Then the tangential velocity $v$ of $P$ is $\bsomega \times \mathbf r$, which has a magnitude $\omega r \sin \theta$ and a direction which is perpendicular to the plane containing $OA$ and $OP$, that is, the plane containing $\bsomega$ and $\mathbf r$.
We also have that in addition to the angular velocity undergone by $B$, let it also be subject to a linear velocity $\mathbf v_0$ in an arbitrary direction.
Then the total velocity $V$ of $P$ is given by:
:$\mathbf V = \mathbf v_0 + \bsomega \times \mathbf r$
and the curl of $V$ is given by:
:$\curl V = \nabla \times \mathbf V = \nabla \times v_0 + \nabla \times \bsomega \times \mathbf r$
Because $v_0$ is constant for all points in $B$, and so is independent of any arbitrary Cartesian coordinates $x$, $y$ and $x$ that may be imposed upon it, $\nabla \times \mathbf v_0$ is zero.
The angular velocity $\bsomega$ is also constant for all points in $B$ at any one time, as $B$ is rigid.
It can be written:
:$\bsomega = \omega_x \mathbf i + \omega_y \mathbf j + \omega_z \mathbf k$
By definition of vector cross product:
:$\bsomega \times \mathbf r = \paren {\omega_y z - \omega_z y} \mathbf i + \paren {\omega_z x - \omega_x z} \mathbf j + \paren {\omega_x y - \omega_y x} \mathbf k$
Because $\omega_x$, $\omega_y$ and $\omega_z$ are not functions of $x$, $y$ and $z$, their partial derivatives $x$, $y$ and $z$ are all zero.
Hence:
== Sources ==
* : Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $5$. Simple Examples of Curl
* 
* 
Category:Curl Operator
Curl Operator Distributes over Addition

== Theorem ==
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions..
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Let $\mathbf f$ and $\mathbf g: \R^3 \to \R^3$ be vector-valued functions on $\R^3$:
:$\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }$
:$\mathbf g := \tuple {\map {g_x} {\mathbf x}, \map {g_y} {\mathbf x}, \map {g_z} {\mathbf x} }$
Let $\nabla \times \mathbf f$ denote the curl of $f$.
Then:
:$\nabla \times \paren {\mathbf f + \mathbf g} = \nabla \times \mathbf f + \nabla \times \mathbf g$
== Proof ==
== Sources ==
* : $\S 22$: Miscellaneous Formulas involving $\nabla$: $22.37$
Category:Curl Operator
Curl Operator on Vector Space is Cross Product of Del Operator

== Theorem ==
Let $R$ be a region of Cartesian $3$-space $\R^3$.
Let $\map {\mathbf V} {x, y, z}$ be a vector field acting over $R$.
Then
:$\curl \mathbf V = \nabla \times \mathbf V$
where:
:$\curl \mathbf V $ denotes the curl of $\mathbf V$
:$\nabla$ denotes the del operator.
== Proof ==
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
We have by definition of curl of $\mathbf V$:
:$\curl \mathbf V = \paren {\dfrac {\partial V_z} {\partial y} - \dfrac {\partial V_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial V_x} {\partial z} - \dfrac {\partial V_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial V_y} {\partial x} - \dfrac {\partial V_x} {\partial y} } \mathbf k$
Now:
== Sources ==
* : Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $4 a$. The Operation $\nabla \times \mathbf V$: $(4.10)$
* : $\S 22$: The Curl: $22.31$
Category:Curl Operator
Curl of Curl is Gradient of Divergence minus Laplacian

== Theorem ==
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.
Let $\mathbf V$ be a vector field on $\R^3$.
Then:
:$\curl \curl \mathbf V = \grad \operatorname {div} \mathbf V - \nabla^2 \mathbf V$
where:
:$\curl$ denotes the curl operator
:$\operatorname {div}$ denotes the divergence operator
:$\grad$ denotes the gradient operator
:$\nabla^2 \mathbf V$ denotes the Laplacian.
 
== Proof ==
From Curl Operator on Vector Space is Cross Product of Del Operator, and Divergence Operator on Vector Space is Dot Product of Del Operator and the definition of the gradient operator:
where $\nabla$ denotes the del operator.
Hence we are to demonstrate that:
:$\nabla \times \paren {\nabla \times \mathbf V} = \map \nabla {\nabla \cdot \mathbf V} - \nabla^2 \mathbf V$
Let $\mathbf V$ be expressed as a vector-valued function on $\mathbf V$:
:$\mathbf V := \tuple {\map {V_x} {\mathbf r}, \map {V_y} {\mathbf r}, \map {V_z} {\mathbf r} }$
where $\mathbf r = \tuple {x, y, z}$ is the position vector of an arbitrary point in $R$.
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
== Also presented as ==
This result can also be presented as:
::$\nabla \times \paren {\nabla \times \mathbf V} = \map \nabla {\nabla \cdot \mathbf V} - \nabla^2 \mathbf V$
presupposing the implementations of $\operatorname {div}$, $\curl$ and $\grad$ as operations using the del operator.
== Sources ==
* : Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $6$. The Operator $\curl \curl$: $(5.7)$, $(5.8)$
* : $\S 22$: Miscellaneous Formulas involving $\nabla$: $22.45$
Category:Gradient Operator
Category:Divergence Operator
Category:Curl Operator
Category:Laplacian
Curl of Gradient is Zero

== Definition ==
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions..
Let $\map U {x, y, z}$ be a scalar field on $\R^3$.
Then:
:$\map \curl {\grad U} = \mathbf 0$
where:
:$\curl$ denotes the curl operator
:$\grad$ denotes the gradient operator.
== Proof ==
From Curl Operator on Vector Space is Cross Product of Del Operator and definition of the gradient operator:
where $\nabla$ denotes the del operator.
Hence we are to demonstrate that:
:$\nabla \times \paren {\nabla U} = \mathbf 0$
Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$.
Then:
From Clairaut's Theorem:
:$\dfrac {\partial^2 U} {\partial x \partial y} = \dfrac {\partial^2 U} {\partial y \partial x}$
and the same for the other partial derivatives.
The result follows.
== Physical Interpretation ==
== Examples ==
== Sources ==
* : Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $2$. The Operator $\curl \grad$: $(5.3)$
* : $\S 22$: Miscellaneous Formulas involving $\nabla$: $22.43$
Category:Gradient Operator
Category:Curl Operator
Category:Curl of Gradient is Zero
Curl of Gradient is Zero/Examples/Electrostatic Field

== Example of Use of Curl of Gradient is Zero ==
Let $R$ be a region of space in which there exists an electric potential field $F$.
From Electric Force is Gradient of Electric Potential Field, the electrostatic force $\mathbf V$ experienced within $R$ is the negative of the gradient of $F$:
:$\mathbf V = -\grad F$
Hence from Curl of Gradient is Zero, the curl of $\mathbf V$ is zero.
== Sources ==
* : Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $2$. The Operator $\curl \grad$
Category:Curl of Gradient is Zero
Category:Electrostatics
Curl of Gradient is Zero/Physical Interpretation

== Physical Interpretation of Curl of Gradient is Zero ==
From Vector Field is Expressible as Gradient of Scalar Field iff Conservative, the vector field given rise to by $\grad F$ is conservative.
The characteristic of a conservative field is that the contour integral around every simple closed contour is zero.
Since the curl is defined as a particular closed contour contour integral, it follows that $\map \curl {\grad F}$ equals zero.
== Sources ==
* : Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $2$. The Operator $\curl \grad$
Category:Curl of Gradient is Zero
Curl of Vector Cross Product

== Definition ==
Let $\R^3 \left({x, y, z}\right)$ denote the real Cartesian space of $3$ dimensions..
Let $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ be the standard ordered basis on $\R^3$.
Let $\mathbf f$ and $\mathbf g: \R^3 \to \R^3$ be vector-valued functions on $\R^3$:
:$\mathbf f := \left({f_x \left({\mathbf x}\right), f_y \left({\mathbf x}\right), f_z \left({\mathbf x}\right)}\right)$
:$\mathbf g := \left({g_x \left({\mathbf x}\right), g_y \left({\mathbf x}\right), g_z \left({\mathbf x}\right)}\right)$
Let $\nabla \times \mathbf f$ denote the curl of $f$.
Then:
:$\nabla \times \left({\mathbf f \times \mathbf g}\right) = \left({\mathbf g \cdot \nabla}\right) \mathbf f - \mathbf g \left({\nabla \cdot \mathbf f}\right) - \left({\mathbf f \cdot \nabla}\right) \mathbf g + \mathbf f \left({\nabla \cdot \mathbf g}\right)$
where:
:$\mathbf f \times \mathbf g$ denotes vector cross product
:$\mathbf f \cdot \nabla$ denotes dot product
:$\nabla \cdot \mathbf f$ denotes divergence.
== Proof ==
== Sources ==
* : $\S 22$: Miscellaneous Formulas involving $\nabla$: $22.41$
Category:Curl Operator
Category:Vector Cross Product
Curl of Vector Field is Solenoidal

== Theorem ==
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.
Let $\mathbf V$ be a vector field on $\R^3$:
Then the curl of $\mathbf V$ is a solenoidal vector field.
== Proof ==
By definition, a solenoidal vector field is one whose divergence is zero.
The result follows from Divergence of Curl is Zero.
== Sources ==
* : Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $5$. The Operator $\operatorname {div} \curl$
Category:Solenoidal Vector Fields
Category:Curl Operator
Current in Electric Circuit/L, R, C in Series

== Theorem ==
Consider the electric circuit $K$ consisting of:
:a resistance $R$
:an inductance $L$
:a capacitance $C$
in series with a source of electromotive force $E$ which is a function of time $t$.
:
The electric current $I$ in $K$ is given by the linear second order ODE:
:$L \dfrac {\d^2 I} {\d t^2} + R \dfrac {\d I} {\d t} + \dfrac 1 C I = \dfrac {\d E} {\d t}$
== Proof ==
Let:
:$E_L$ be the drop in electromotive force across $L$
:$E_R$ be the drop in electromotive force across $R$
:$E_C$ be the drop in electromotive force across $C$.
From Kirchhoff's Voltage Law:
:$E - E_L - E_R - E_C = 0$
From Ohm's Law:
:$E_R = R I$
From Drop in EMF caused by Inductance is proportional to Rate of Change of Current:
:$E_L = L \dfrac {\d I} {\d t}$
From Drop in EMF caused by Capacitance is proportional to Accumulated Charge:
:$E_C = \dfrac 1 C Q$
where $Q$ is the electric charge $Q$ that has accumulated on $C$.
Thus:
:$E - L \dfrac {\d I} {\d t} - R I - \dfrac 1 C Q = 0$
which can be rewritten:
:$L \dfrac {\d I} {\d t} + R I + \dfrac 1 C Q = E$
Differentiating $t$ gives:
:$L \dfrac {\d^2 I} {\d t^2} + R \dfrac {\d I} {\d t} + \dfrac 1 C I = \dfrac {\d E} {\d t}$
== Sources ==
* : $\S 2.13$: Simple Electric Circuits
Category:Electronics
Current in Electric Circuit/L, R in Series

== Theorem ==
Consider the electric circuit $K$ consisting of:
:a resistance $R$
:an inductance $L$
in series with a source of electromotive force $E$ which is a function of time $t$.
:
The electric current $I$ in $K$ is given by the linear first order ODE:
:$L \dfrac {\d I} {\d t} + R I = E$
=== Constant EMF at $t = 0$ ===
=== Exponentially Decaying EMF at $t = 0$ ===
=== Sinusoidal EMF ===
=== Condition for Ohm's Law ===
=== Minimum Current implies Increasing EMF ===
=== Maximum Current implies Decreasing EMF ===
== Proof ==
Let:
:$E_L$ be the drop in electromotive force across $L$
:$E_R$ be the drop in electromotive force across $R$
From Kirchhoff's Voltage Law:
:$E - E_L - E_R = 0$
From Ohm's Law:
:$E_R = R I$
From Drop in EMF caused by Inductance is proportional to Rate of Change of Current:
:$E_L = L \dfrac {\d I} {\d t}$
Thus:
:$E - L \dfrac {\d I} {\d t} - R I = 0$
which can be rewritten:
:$L \dfrac {\d I} {\d t} + R I = E$
== Sources ==
* : $\S 2.13$: Simple Electric Circuits
Category:Electronics
Category:Decay Equation
Current in Electric Circuit/L, R in Series/Condition for Ohm's Law

== Theorem ==
Consider the electric circuit $K$ consisting of:
:a resistance $R$
:an inductance $L$
in series with a source of electromotive force $E$ which is a function of time $t$.
:
Ohm's Law is satisfied by $K$ whenever the current $I$ is at a maximum or a minimum.
== Proof ==
From Electric Current in Electric Circuit: L, R in Series:
:$L \dfrac {\d I} {\d t} + R I = E$
defines the behaviour of $I$.
Let $I$ be at a maximum or a minimum.
Then from Derivative at Maximum or Minimum:
:$\dfrac {\d I} {\d t} = 0$
and so:
which is Ohm's Law.
== Sources ==
* : $\S 2.13$: Simple Electric Circuits: Problem $2 \ \text{(a)}$
Category:Electronics
Current in Electric Circuit/L, R in Series/Constant EMF at t = 0

== Theorem ==
Consider the electric circuit $K$ consisting of:
:a resistance $R$
:an inductance $L$
in series with a source of electromotive force $E$ which is a function of time $t$.
:
Let the electric current flowing in $K$ at time $t = 0$ be $I_0$.
Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$.
The electric current $I$ in $K$ is given by the equation:
:$I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$
=== Corollary 1 ===
=== Corollary 2 ===
=== Corollary 3 ===
== Proof ==
From Electric Current in Electric Circuit: L, R in Series:
:$L \dfrac {\d I} {\d t} + R I = E_0$
defines the behaviour of $I$.
This can be written as:
:$(1): \quad \dfrac {\d I} {\d t} = \dfrac R L \paren {\dfrac {E_0} R - I}$
$(1)$ is in the form:
:$\dfrac {\d y} {\d x} = k \paren {y_a - y}$
where:
:$k \in \R: k > 0$
:$y = y_0$ at $x = 0$
This is an example of the Decay Equation, where:
:$k = \dfrac R L$
:$y_a = \dfrac {E_0} R$
:$y_0 = I_0$.
whose particular solution is:
:$y = y_a + \paren {y_0 - y_a} e^{-k x}$
Hence the particular solution to $(1)$ is:
:$I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$
== Sources ==
* : $\S 2.13$: Simple Electric Circuits: Example $1$
Category:Electronics
Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 1

== Theorem ==
Consider the electric circuit $K$ consisting of:
:a resistance $R$
:an inductance $L$
in series with a source of electromotive force $E$ which is a function of time $t$.
:
Let the electric current flowing in $K$ at time $t = 0$ be $I_0$.
Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$.
After a sufficiently long time, the electric current $I$ in $K$ is given by the equation:
:$E_0 = R I$
== Proof ==
From Electric Current in Electric Circuit: L, R in Series: Constant EMF at $t = 0$:
:$I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$
We have that:
:$\ds \lim_{t \mathop \to \infty} e^{-R t / L} \to 0$
and so:
:$\ds \lim_{t \mathop \to \infty} I \to \dfrac {E_0} R$
Hence the result.
== Sources ==
* : $\S 2.13$: Simple Electric Circuits: Example $1$
Category:Electronics