Added signal analysis to mathematical physics section.
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- 'Second order differential equations': mathematics/ordinary-differential-equations/second-order-ode.md
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- 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md
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- 'The Laplace transform': mathematics/ordinary-differential-equations/laplace-transform.md
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- 'Mathematical physics':
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- 'Signal analysis':
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- 'Signals': mathematics/mathematical-physics/signal-analysis/signals.md
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- 'Physics':
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- physics/index.md
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# Signals
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## Definitions
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> *Definition*: a signal is a function of space and time.
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> * Output can be analog or quantised.
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> * Input can be continuous or discrete.
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<br>
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> *Definition*: a signal can be sampled at particular moments $k T_s$ in time, with $k \in \mathbb{Z}$ and $T_s \in \mathbb{R}$ the sampling period. For a signal $f: \mathbb{R} \to \mathbb{R}$ sampled with a sampling period $T_s$ may be denoted by
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>
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> $$
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> f[k] = f(kT_s), \qquad \forall k \in \mathbb{Z}.
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> $$
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<br>
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> *Definition*: signal transformations on a function $x: \mathbb{R} \to \mathbb{R}$ obtaining the function $y: \mathbb{R} \to \mathbb{R}$ are given by
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>
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> | Signal transformation | Time | Amplitude |
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> | :-: | :-: | :-: |
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> | Reversal | $y(t) = x(-t)$ | $y(t) = -x(t)$ |
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> | Scaling | $y(t) = x(at)$ | $y(t) = ax(t)$ |
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> | Shifting | $y(t) = x(t - b)$ | $y(t) = x(t) + b$ |
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>
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> for all $t \in \mathbb{R}$.
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For sampled signals similar definitions hold.
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### Symmetry
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> *Definition*: consider a signal $f: \mathbb{R} \to \mathbb{R}$ which is defined in an interval which is symmetric around $t = 0$, we define.
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>
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> * $f$ is *even* if $f(t) = f(-t)$, $\forall t \in \mathbb{R}$.
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> * $f$ is *odd* if $f(t) = -f(-t)$, $\forall t \in \mathbb{R}$.
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For sampled signals similar definitions hold.
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> *Theorem*: every signal can be decomposed into symmetric parts.
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??? note "*Proof*:"
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Will be added later.
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### Periodicity
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> *Definition*: a signal $f: \mathbb{R} \to \mathbb{R}$ is defined to be periodic in $T$ if and only if
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>
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> $$
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> f(t + T) = f(t), \qquad \forall t \in \mathbb{R}.
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> $$
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For sampled signals similar definitions hold.
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> *Theorem*: a summation of two periodic signals with periods $T_1, T_2 \in \mathbb{R}$ respectively is periodic if and only if
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>
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> $$
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> \frac{T_1}{T_2} \in \mathbb{Q}.
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> $$
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??? note "*Proof*:"
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Will be added later.
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### Signals
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> *Definition*: the Heaviside step signal $u: \mathbb{R} \to \mathbb{R}$ is defined by
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>
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> $$
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> u(t) = \begin{cases} 1 &\text{ if } t > 0,\\ 0 &\text{ if } t < 0,\end{cases}
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> $$
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>
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> for all $t \in \mathbb{R}$.
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For a sampled function the Heaviside step signal is given by
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$$
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u[k] = \begin{cases} 1 \text{ if } k \geq 0, \\ 0 \text{ if } k < 0, \end{cases}
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$$
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for all $k \in \mathbb{Z}$.
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> *Definition*: the rectangular signal $\text{rect}: \mathbb{R} \to \mathbb{R}$ is defined by
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>
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> $$
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> \text{rect} (t) = \begin{cases} 1 &\text{ if } |t| < \frac{1}{2}, \\ 0 &\text{ if } |t| > \frac{1}{2},\end{cases}
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> $$
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>
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> for all $t \in \mathbb{R}$.
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The rect signal can be normalised obtaining the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ defined by
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$$
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D(t, \varepsilon) = \begin{cases} \frac{1}{\varepsilon} &\text{ if } |t| < \frac{\varepsilon}{2},\\ 0 &\text{ if } |t| > \frac{\varepsilon}{2},\end{cases}
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$$
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for all $t \in \mathbb{R}$.
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The following signal has been derived from the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ used on a signal $f: \mathbb{R} \to \mathbb{R}$ for
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$$
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\lim_{\varepsilon \;\downarrow\; 0} \int_{-\infty}^{\infty} f(t) D(t, \varepsilon)dt = \lim_{\varepsilon \;\downarrow\; 0} \frac{1}{\varepsilon} \int_{-\frac{\varepsilon}{2}}^{\frac{\varepsilon}{2}} f(t) dt = f(0),
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$$
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using the mean [value theorem for integrals](../calculus/integration.md#the-mean-value-theorem-for-integrals).
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> *Definition*: the Dirac signal $\delta$ is a generalized signal defined by the properties
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>
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> $$
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> \begin{align*}
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> \delta(t - t_0) = 0 \quad \text{ for } t \neq t_0,& \\
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> \int_{-\infty}^\infty f(t) \delta(t - t_0) dt = f(t_0),&
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> \end{align*}
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> $$
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>
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> for a signal $f: \mathbb{R} \to \mathbb{R}$ continuous in $t_0$.
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For sampled signals the $\delta$ signal is given by
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$$
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\delta[k] = \begin{cases} 1 &\text{ if } k = 0, \\ 0 &\text{ if } k \neq 0.\end{cases}
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$$
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## Signal sampling
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We already established that a signal $f: \mathbb{R} \to \mathbb{R}$ can be sampled with a sampling period $T_s \in \mathbb{R}$ obtaining $f[k] = f(kT_s)$ for all $k \in \mathbb{Z}$. We can also define a *time-continuous* signal $f_s(t)$ that represents the sampled signal using the Dirac signal, obtaining
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$$
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f_s(t) = f(t) \sum_{k = - \infty}^\infty \delta(t - k T_s), \qquad \forall t \in \mathbb{R}.
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$$
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> *Definition*: the sampling signal or impulse train $\delta_{T_s}: \mathbb{R} \to \mathbb{R}$ is defined as
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>
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> $$
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> \delta_{T_s}(t) = \sum_{k = - \infty}^\infty \delta(t - k T_s)
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> $$
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>
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> for all $t \in \mathbb{R}$ with a sampling period $T_s \in \mathbb{R}$.
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Then integration works out since we have
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$$
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\int_{-\infty}^\infty f(t) \delta_{T_s}(t) dt = \sum_{k = -\infty}^\infty \int_{-\infty}^\infty f(t) \delta(t - k T_s) dt = \sum_{k = -\infty}^\infty f [k],
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$$
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by definition.
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## Convolutions
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> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the convolution product is defined as
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>
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> $$
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> f(t) * g(t) = \int_{-\infty}^\infty f(u)g(t-u)du
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> $$
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>
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> for all $t \in \mathbb{R}$.
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<br>
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> *Proposition*: the convolution product is commutative, distributive and associative.
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??? note "*Proof*:"
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Will be added later.
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> *Theorem*: let $f: \mathbb{R} \to \mathbb{R}$ be a signal then we have for the convolution product between $f$ and the Dirac signal $\delta$ and some $t_0 \in \mathbb{R}$
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>
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> $$
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> f(t) * \delta(t - t_0) = f(t - t_0)
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> $$
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>
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> for all $t \in \mathbb{R}$
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??? note "*Proof*:"
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let $f: \mathbb{R} \to \mathbb{R}$ be a signal and $t_0 \in \mathbb{R}$, using the definition of the Dirac signal
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$$
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f(t) * \delta(t - t_0) = \int_{-\infty}^\infty f(u) \delta(t - t_0 - u)du = f(t - t_0),
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$$
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for all $t \in \mathbb{R}$.
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In particular $f(t) * \delta(t) = f(t)$ for all $t \in \mathbb{R}$; $\delta$ is the unity of the convolution.
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The average value of a signal $f: \mathbb{R} \to \mathbb{R}$ for an interval $\varepsilon \in \mathbb{R}$ may be given by
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$$
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f(t) * D(t, \varepsilon) = \frac{1}{\varepsilon} \int_{t - \frac{\varepsilon}{2}}^{t + \frac{\varepsilon}{2}} f(u)du.
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$$
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For sampled/discrete signals we have a similar definition for the convolution product, given by
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$$
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f[k] * g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k - m],
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$$
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for all $k \in \mathbb{Z}$.
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## Correlations
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> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the cross-correlation is defined as
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>
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> $$
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> f(t) \star g(t) = \int_{-\infty}^\infty f(u) g(t + u)du
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> $$
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>
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> for all $t \in \mathbb{R}$.
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Especially the auto-correlation of a continuous signal $f: \mathbb{R} \to \mathbb{R}$ given by $f(t) \star f(t)$ for all $t \in \mathbb{R}$ is useful, as it can detect periodicity without stating the proof.
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For sampled/discrete signals a similar definition exists given by
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$$
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f[k] \star g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k + m],
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$$
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for all $k \in \mathbb{Z}$.
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@ -1,3 +1,14 @@
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# Physics
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Welcome to the physics page.
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Welcome to the physics page. Some special physical environments that will be used in this seection are listed and explained below.
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* *Principles*: a fundamental rule or concept in physics serving as a basis for reasoning.
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* *Assumptions*: a less fundamental rule or concept in physics that is taken to be true such that certain phenoma can be simplified.
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* *Definitions*: a precise and unambiguous description of the meaning of a physical term. It characterizes the meaning of a word by giving all the properties an only those properties that must be true.
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* *Laws*: a physical statement that is proved to be true, under the made assumptions and posed principles. In a physical text, the term law is often reserved for the most important results.
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* *Propositions*: an often interesting result, but generally less important than a law.
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* *Lemmas*: a minor result whose purpose is to help in proving a law. It is a stepping stone on the path to proving a law.
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* *Corollaries*: a result in which the proof relies heavily on a given law.
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* *Proofs*: a convincing argument that a certain physical statement is necessarily true. A proof generally uses deductive reasoning and logic but also contains some amount of ordinary language.
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The physics sections of this wiki are based on various books and lectures. A comprehensive list of references can be found below.
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