Applied Mathematics/Printable version

Applied Mathematics

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The Basics of Theory of The Fourier Transform

The Basics

The two most important things in Theory of The Fourier Transform are "differential calculus" and "integral calculus". The readers are required to learn "differential calculus" and "integral calculus" before studying the Theory of The Fourier Transform. Hence, we will learn them on this page.

Differential calculus

The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

Differentiation is the process of finding a derivative of the function $f(x)$ in the independent input x. The differentiation of $f(x)$ is denoted as $f'(x)$ or ${\frac {d}{dx}}f(x)$. Both of the two notations are same meaning.

Differentiation is manipulated as follows:
${\frac {d}{dx}}(x^{3}+1)$
$=3x^{3-1}$
$=3x^{2}$

As you see, in differentiation, the number of the degree of the variable is multiplied to the variable, while the degree is subtracted one from itself at the same time. The term which doesn't have the variable $x$ is just removed in differentiation.

Examples

${\frac {d}{dx}}(x^{5}+x^{2}+28)$
$=5x^{5-1}+2x^{2-1}$ 28 doesn't have the variable x, so 28 is removed
$=5x^{4}+2x^{1}$
$=5x^{4}+2x$

${\frac {d}{dx}}(x^{7}+x^{4}+x+7)$
$=7x^{7-1}+4x^{4-1}+1x^{1-1}$ 7 doesn't have the variable x, so 7 is removed
$=7x^{6}+4x^{3}++1x^{0}$
$=7x^{6}+4x^{3}+1$

Practice problems

(1)${\frac {d}{dx}}(x^{4}+15)=$

(2)${\frac {d}{dx}}(x^{5}+x^{3}+x)=$

Integral calculus

If you differentiate $f(x)=x^{3}$ or $f(x)=x^{3}-2$, each of them become $f'(x)=3x^{2}$. Then let's think of the opposite case. A function is provided, and when the function is differentiated, the function became $f'(x)=3x^{2}$. What's the original function? To find the original function, the integral calculus is used. Integration of $f(x)$ is denoted as $\int f(x)dx$.

Integration is manipulated as follows:
$\int 3x^{2}dx$
$={\frac {3x^{2+1}}{2+1}}+C$
$=x^{3}+C$
$C$ denotes Constant in the equation.

More generally speaking, the integration of f(x) is defined as:

$\int x^{n}dx={\frac {x^{n+1}}{n+1}}+C$

Definite integral

Definite integral is defined as follows:
$\int _{a}^{b}f(x)dx$
$=[F(x)]_{a}^{b}$
$=F(b)-F(a)$
where $F(x)=\int f(x)dx$

Examples

(1)$\int 4xdx$
$={\frac {4x^{1+1}}{1+1}}+C$
$=2x^{2}+C$

(2)$\int _{1}^{2}(2x+1)dx$
$=[x^{2}+x]_{1}^{2}$
$=(4+2)-(1+1)$
$=4$

Practice problems

(1)$\int 6xdx=$
(2)$\int _{0}^{2}(3x^{2}+3)dx=$

Euler's number "e"

Euler's number $e$ (also known as Napier's constant) has special features in differentiation and integration:

$\int e^{x}dx=e^{x}+C$

${\frac {d}{dx}}e^{x}=e^{x}$

By the way, in Mathematics, $exp(x)$ denotes $e^{x}$.

Fourier Series

For the function $f(x)$, Taylor expansion is possible.

$f(x)=f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}+\cdots$

This is the Taylor expansion of $f(x)$. On the other hand, more generally speaking, $f(x)$ can be expanded by also Orthogonal f

Fourier series

For the function $f(x)$ which has $2\pi$ for its period, the series below is defined:

${\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cos nx+b_{n}\sin nx)\cdots (1)$

This series is referred to as Fourier series of $f(x)$. $a_{n}$ and $b_{n}$ are called Fourier coefficients.

$a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\cos(nx)dx$

$b_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\sin(nx)dx$

where $n$ is natural number. Especially when the Fourier series is equal to the $f(x)$, (1) is called Fourier series expansion of $f(x)$. Thus Fourier series expansion is defined as follows:

$f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }(a_{n}\cos nx+b_{n}\sin nx)$

General Fourier Transform

Fourier Transform

The Fourier transform relates the function's time domain, shown in red, to the function's frequency domain, shown in blue. The component frequencies, spread across the frequency spectrum, are represented as peaks in the frequency domain.

Fourier Transform is to transform the function which has certain kinds of variables, such as time or spatial coordinate, $f(t)$ for example, to the function which has variable of frequency.

Definition

${\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(t)\ e^{-i2\pi t\xi }\,dt$...(1)

This integral above is referred to as Fourier integral, while ${\hat {f}}(\xi )$ is called Fourier transform of $f(t)$. $t$ denotes "time". $\xi$ denotes "frequency".

On the other hand, Inverse Fourier transform is defined as follows:

$f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )\ e^{i2\pi t\xi }\,d\xi$ ...(2)

In the textbooks of universities, the Fourier transform is usually introduced with the variable Angular frequency $\omega$. In other word, $\xi \rightarrow \omega =2\pi \xi$ is substituted to (1) and (2) in the books. In that case, the Fourier transform is written in two different ways.

1.

${\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega t}dt$

$f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}(t)e^{i\omega t}d\omega$

2.

${\hat {f}}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)e^{-i\omega t}dt$

$f(t)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {f}}(t)e^{i\omega t}d\omega$

Fourier Sine Series

Fourier sine series

The series below is called Fourier sine series.

$f(x)=\sum _{n=1}^{\infty }b_{n}\sin {\frac {n\pi x}{L}}$

where

$b_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin {\frac {n\pi x}{L}}dx$

Fourier Cosine Series

Fourier cosine series

The series below is called Fourier cosine series.

$f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }a_{n}\cos {\frac {n\pi x}{L}}$

where

$a_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\cos {\frac {n\pi x}{L}}dx$

Fourier Integral Transforms

Let $f(x)=(-\infty ,\infty )$ and

suppose

$\int _{-\infty }^{\infty }|f(t)|dt\leq M$.

Then we have the functions below.

$f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}(\omega )e^{i\omega t}d\omega$

This function $f(t)$ is referred to as Fourier integral.

${\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega t}dt$

This function ${\hat {f}}(\omega )$ is referred to as fourier transform as we previously learned.

Parseval's Theorem

Parseval's theorem

$\int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }|X(f)|^{2}\,df$

where $X(f)={\mathcal {F}}\{x(t)\}$ represents the continuous Fourier transform of x(t) and f represents the frequency component of x. The function above is called Parseval's theorem.

Derivation

Let ${\bar {X}}(f)$ be the complex conjugation of $X(f)$.

$X(-f)=\int _{-\infty }^{\infty }x(-t)e^{-ift}$

$=\int _{-\infty }^{\infty }x(t)e^{ift}$

$={\bar {X}}(f)$

$\int _{-\infty }^{\infty }|X(f)|^{2}\,df$

Here, we know that $X(f)$ is equal to the expansion coefficient of $x(t)$ in fourier transforming of $x(t)$.
Hence, the integral of $|X(f)|^{2}$ is

$\int _{-\infty }^{\infty }{\bar {X}}(f)X(f)\,df$

$=\int _{-\infty }^{\infty }\left({\frac {1}{{\sqrt {2}}\pi }}\int _{-\infty }^{\infty }x(t)e^{ift}dt\right)\left({\frac {1}{{\sqrt {2}}\pi }}\int _{-\infty }^{\infty }x(t')e^{-ift'}dt'\right)df$

$=\int _{-\infty }^{\infty }x(t)x(t')\left({\frac {1}{2\pi }}e^{-if(t-t')}df\right)dtdt'$

$=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x(t)x(t')\delta (t-t')dtdt'$

$=\int _{-\infty }^{\infty }|x(t)|^{2}dt$

Hence

$\int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }|X(f)|^{2}\,df$

Bessel Functions

Bessel Functions

The equation below is called Bessel's differential equation.

$x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+(x^{2}-n^{2})y=0$

The two distinctive solutions of Bessel's differential equation are either one of the two pairs: (1)Linear combination of Bessel function(also known as Bessel function of the first kind) and Neumann function(also known as Bessel function of the second kind) (2)Linear combination of Hankel function of the first kind and Hankel function of the second kind. Bessel function (of the first kind) is denoted as $\displaystyle J_{n}(x)$. Bessel function is defined as follow:

$J_{n}(x)={\frac {x^{n}}{2^{n}\Gamma (1-n)}}(1-{\frac {x^{2}}{2(2n+2)}}+{\frac {x^{4}}{2\cdot 4(2n+2)(2n+4)}}-\cdots )$

$=\sum _{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+n+1)}}{\left({\frac {x}{2}}\right)}^{2m+n}$

where

$H_{n}^{(1)}(x)=J_{n}(x)+iN_{n}(x)$

$H_{n}^{(2)}(x)=J_{n}(x)-iN_{n}(x)$

$N_{n}(x)={\frac {J_{n}(x)\cos(n\pi )-J_{-n}(x)}{\sin(n\pi )}}$

Γ(z) is the gamma function. $i$ is the imaginary unit. $N_{n}(x)$ is the Neumann function(or Bessel function of the second kind). $H_{n}(x)$ is the Hankel functions. If n is an integer, the Bessel function of the first kind is an entire function.

Laplace Transforms

The Laplace transform is an integral transform which is widely used in physics and engineering.

Laplace Transforms involve a technique to change an expression into another form that is easier to work with using an improper integral. We usually introduce Laplace Transforms in the context of differential equations, since we use them a lot to solve some differential equations that can't be solved using other standard techniques. However, Laplace Transforms require only improper integration techniques to use. So you may run across them in first year calculus.

Notation: The Laplace Transform is denoted as $\displaystyle {\mathcal {L}}\left\{f(t)\right\}$.

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace.

Topics You Need to Understand For This Page Improper Integrals

Definition

For a function $f(t)$, using Napier's constant $e$ and a complex number $s$, the Laplace transform $F(s)$ is defined as follows:

$F(s)={\mathcal {L}}\left\{f(t)\right\}(s)=\int _{0}^{\infty }e^{-st}f(t)\,dt$

The parameter $s$ is a complex number.

$s=\sigma +i\omega ,\,$ with real numbers $\sigma$ and $\omega$.

This $F(s)$ is the Laplace transform of $f(t)$.

Explanation

Here is what is going on.

Examples of Laplace transform

Examples of Laplace transform

$f(t)$	$F(s)={\mathcal {L}}\{f(t)\}$
$C$	${\frac {C}{s}}$
$t$	${\frac {1}{s^{2}}}$
$t^{n}$	${\frac {n!}{s^{n+1}}}$
${\frac {t^{n-1}}{(n-1)!}}$	${\frac {1}{s^{n}}}$
$e^{at}$	${\frac {1}{s-a}}$
$e^{-at}$	${\frac {1}{s+a}}$
${\rm {cos}}\ \omega t$	${\frac {s}{s^{2}+{\omega }^{2}}}$
${\rm {sin}}\ \omega t$	${\frac {\omega }{s^{2}+{\omega }^{2}}}$
${\frac {t^{n-1}}{\Gamma (n)}}$	${\frac {1}{s^{n}}}$ (n>0)
$\delta (t-a)$	$e^{-as}$
$H(t-a)$	${\frac {e^{-as}}{s}}$

In the above table,

1. $C$ and $a$ are constants
2. $n$ is a natural number
3. $\delta (t-a)$ is the Delta function
4. $H(t-a)$ is the Heaviside function

ID	Function	Time domain $x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}$	Laplace domain $X(s)={\mathcal {L}}\left\{x(t)\right\}$	Region of convergence for causal systems
1	Ideal delay	$\delta (t-\tau )\$	$e^{-\tau s}\$
1a	Unit impulse	$\delta (t)\$	$1\$	$\mathrm {all} \ s\,$
2	Delayed nth power with frequency shift	${\frac {(t-\tau )^{n}}{n!}}e^{-\alpha (t-\tau )}\cdot u(t-\tau )$	${\frac {e^{-\tau s}}{(s+\alpha )^{n+1}}}$	$s>0\,$
2a	nth Power	${t^{n} \over n!}\cdot u(t)$	${1 \over s^{n+1}}$	$s>0\,$
2a.1	qth Power	${t^{q} \over \Gamma (q+1)}\cdot u(t)$	${1 \over s^{q+1}}$	$s>0\,$
2a.2	Unit step	$u(t)\$	${1 \over s}$	$s>0\,$
2b	Delayed unit step	$u(t-\tau )\$	${e^{-\tau s} \over s}$	$s>0\,$
2c	Ramp	$t\cdot u(t)\$	${\frac {1}{s^{2}}}$	$s>0\,$
2d	nth Power with frequency shift	${\frac {t^{n}}{n!}}e^{-\alpha t}\cdot u(t)$	${\frac {1}{(s+\alpha )^{n+1}}}$	$s>-\alpha \,$
2d.1	Exponential decay	$e^{-\alpha t}\cdot u(t)\$	${1 \over s+\alpha }$	$s>-\alpha \$
3	Exponential approach	$(1-e^{-\alpha t})\cdot u(t)\$	${\frac {\alpha }{s(s+\alpha )}}$	$s>0\$
4	Sine	$\sin(\omega t)\cdot u(t)\$	${\omega \over s^{2}+\omega ^{2}}$	$s>0\$
5	Cosine	$\cos(\omega t)\cdot u(t)\$	${s \over s^{2}+\omega ^{2}}$	$s>0\$
6	Hyperbolic sine	$\sinh(\alpha t)\cdot u(t)\$	${\alpha \over s^{2}-\alpha ^{2}}$	$s>|\alpha |\$
7	Hyperbolic cosine	$\cosh(\alpha t)\cdot u(t)\$	${s \over s^{2}-\alpha ^{2}}$	$s>|\alpha |\$
8	Exponentially-decaying sine	$e^{-\alpha t}\sin(\omega t)\cdot u(t)\$	${\omega \over (s+\alpha )^{2}+\omega ^{2}}$	$s>-\alpha \$
9	Exponentially-decaying cosine	$e^{-\alpha t}\cos(\omega t)\cdot u(t)\$	${s+\alpha \over (s+\alpha )^{2}+\omega ^{2}}$	$s>-\alpha \$
10	nth Root	${\sqrt[{n}]{t}}\cdot u(t)$	$s^{-(n+1)/n}\cdot \Gamma \left(1+{\frac {1}{n}}\right)$	$s>0\,$
11	Natural logarithm	$\ln \left({t \over t_{0}}\right)\cdot u(t)$	$-{t_{0} \over s}\ [\ \ln(t_{0}s)+\gamma \ ]$	$s>0\,$
12	Bessel function of the first kind, of order n	$J_{n}(\omega t)\cdot u(t)$	${\frac {\omega ^{n}\left(s+{\sqrt {s^{2}+\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}+\omega ^{2}}}}$	$s>0\,$ $(n>-1)\,$
13	Modified Bessel function of the first kind, of order n	$I_{n}(\omega t)\cdot u(t)$	${\frac {\omega ^{n}\left(s+{\sqrt {s^{2}-\omega ^{2}}}\right)^{-n}}{\sqrt {s^{2}-\omega ^{2}}}}$	$s>|\omega |\,$
14	Bessel function of the second kind, of order 0	$Y_{0}(\alpha t)\cdot u(t)$
15	Modified Bessel function of the second kind, of order 0	$K_{0}(\alpha t)\cdot u(t)$
16	Error function	$\mathrm {erf} (t)\cdot u(t)$	${e^{s^{2}/4}\operatorname {erfc} \left(s/2\right) \over s}$	$s>0\,$
17	Constant	$C$	$\displaystyle {\frac {C}{s}}$
Explanatory notes: $u(t)\,$ represents the Heaviside step function. $\delta (t)\,$ represents the Dirac delta function. $\Gamma (z)\,$ represents the Gamma function. $\gamma \,$ is the Euler-Mascheroni constant. $t\,$, a real number, typically represents time, although it can represent any independent dimension. $s\,$ is the complex angular frequency. $\alpha \,$, $\beta \,$, $\tau \,$, $\omega \,$ and $C$ are real numbers. $n\,$is an integer. A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the ROC for causal systems is not the same as the ROC for anticausal systems. See also causality.

Examples

1. Calculate ${\mathcal {L}}\{C\}$ (where $C$ is a constant) using the integral definition.

${\begin{array}{rcl}\displaystyle {\int _{0}^{\infty }e^{-st}C\,dt}&=&C\displaystyle {\int _{0}^{\infty }e^{-st}\,dt}\\&=&\displaystyle {C\lim _{b\to \infty }{\int _{0}^{b}e^{-st}\,dt}}\\&=&\displaystyle {\left.C\lim _{b\to \infty }{\frac {e^{-st}}{-s}}\right|_{t=0}^{t=b}}\\&=&\displaystyle {-{\frac {C}{s}}\left[\lim _{b\to \infty }{e^{-bs}}-e^{0}\right]}\\&=&\displaystyle {-{\frac {C}{s}}[0-1]}\\&=&\displaystyle {\frac {C}{s}}\end{array}}$

$\therefore \displaystyle {\mathcal {L}}\left\{C\right\}={\frac {C}{s}}$

---

2. Calculate ${\mathcal {L}}\{e^{-at}\}$ using the integral definition.

${\begin{array}{rcl}\displaystyle \int _{0}^{\infty }e^{-st}\cdot e^{-at}\,dt&=&\displaystyle \int _{0}^{\infty }e^{-(s+a)t}\,dt\\&=&\displaystyle \lim _{b\to \infty }{\displaystyle \int _{0}^{b}e^{-(s+a)t}\,dt}\\&=&\displaystyle \lim _{b\to \infty }\left[\displaystyle {\frac {-e^{-(s+a)t}}{s+a}}\right]_{t=0}^{t=b}\\&=&\displaystyle \lim _{b\to \infty }\left[\displaystyle {\frac {-e^{-(s+a)b}}{s+a}}-\displaystyle {\frac {-e^{-(s+a)0}}{s+a}}\right]\\&=&\displaystyle {\frac {1}{s+a}}\end{array}}$

$\therefore {\mathcal {L}}\{e^{-at}\}=\displaystyle {\frac {1}{s+a}}$

Complex Integration

Complex integration

On the piecewise smooth curve $C:z=z(t)$ $(a\leqq t\leqq b)$, suppose the function f(z) is continuous. Then we obtain the equation below.

$\int _{C}f(z)dz=\int _{a}^{b}f\{z(t)\}\ {\frac {dz(t)}{dt}}dt$

where $f(z)$ is the complex function, and $z$ is the complex variable.

Proof

Let

$f(z)=u(x,y)+iv(x,y)$

$dz=dx+idy$

Then

$\int _{C}f(z)dz$

$=\int _{C}(u+iv)(dx+idy)$

$=\left(\int _{C}udx-\int _{C}vdy\right)+i\left(\int _{C}vdx-\int _{C}udy\right)$

The right side of the equation is the real integral, therefore, according to calculus, the relationship below can be applied.

$\int _{x_{1}}^{x_{2}}f(x)dx=\int _{t_{1}}^{t_{2}}f(x){\frac {dx}{dt}}dt$

Hence

$\int _{C}f(z)dz$

$=\left(\int _{a}^{b}u{\frac {dx}{dt}}dt-\int _{a}^{b}v{\frac {dy}{dt}}dt\right)+i\left(\int _{a}^{b}v{\frac {dx}{dt}}dt-\int _{a}^{b}u{\frac {dy}{dt}}dt\right)$

$=\left(\int _{a}^{b}ux'(t)dt-\int _{a}^{b}vy'(t)dt\right)+i\left(\int _{a}^{b}vx'(t)dt-\int _{a}^{b}uy'(t)dt\right)$

$=(u+iv)\left(x'(t)+iy'(t)\right)dt$

$=\int _{a}^{b}f\{z(t)\}\ z'(t)dt$

This completes the proof.

The Basics

The Basics of linear algebra

$\mathbf {A} ={\begin{bmatrix}a_{11}&a_{12}&a_{12}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\\end{bmatrix}}.$

A matrix is composed of a rectangular array of numbers arranged in rows and columns. The horizontal lines are called rows and the vertical lines are called columns. The individual items in a matrix are called elements. The element in the i-th row and the j-th column of a matrix is referred to as the i,j, (i,j), or (i,j)th element of the matrix. To specify the size of a matrix, a matrix with m rows and n columns is called an m-by-n matrix, and m and n are called its dimensions.

Basic operation

Operation	Definition	Example
Addition	The sum A+B of two m-by-n matrices A and B is calculated entrywise: (A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n.	${\begin{bmatrix}1&3&1\\1&0&0\end{bmatrix}}+{\begin{bmatrix}0&0&5\\7&5&0\end{bmatrix}}={\begin{bmatrix}1+0&3+0&1+5\\1+7&0+5&0+0\end{bmatrix}}={\begin{bmatrix}1&3&6\\8&5&0\end{bmatrix}}$
Scalar multiplication	The scalar multiplication cA of a matrix A and a number c (also called a scalar in the parlance of abstract algebra) is given by multiplying every entry of A by c: (cA)i,j = c · Ai,j.	$2\cdot {\begin{bmatrix}1&8&-3\\4&-2&5\end{bmatrix}}={\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{bmatrix}}={\begin{bmatrix}2&16&-6\\8&-4&10\end{bmatrix}}$
Transpose	The transpose of an m-by-n matrix A is the n-by-m matrix AT (also denoted Atr or tA) formed by turning rows into columns and vice versa: (AT)i,j = Aj,i.	${\begin{bmatrix}1&2&3\\0&-6&7\end{bmatrix}}^{\mathrm {T} }={\begin{bmatrix}1&0\\2&-6\\3&7\end{bmatrix}}$

Practice problems

(1) ${\begin{bmatrix}5&7&3\\1&2&9\end{bmatrix}}+{\begin{bmatrix}4&0&5\\8&3&0\end{bmatrix}}=$
(2) $4{\begin{bmatrix}-1&0&-5\\7&9&-6\end{bmatrix}}=$
(3) ${\begin{bmatrix}-2&5&7\\0&0&9\end{bmatrix}}^{\mathrm {T} }=$

Matrix multiplication

Multiplication of two matrices is defined only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B

$[\mathbf {AB} ]_{i,j}=A_{i,1}B_{1,j}+A_{i,2}B_{2,j}+\cdots +A_{i,n}B_{n,j}=\sum _{r=1}^{n}A_{i,r}B_{r,j}$

Schematic depiction of the matrix product AB of two matrices A and B.

Example

${\begin{bmatrix}-2&0\\3&2\end{bmatrix}}{\begin{bmatrix}1&2\\3&-1\end{bmatrix}}$

$={\begin{bmatrix}-2+0&-4+0\\3+6&6+(-2)\end{bmatrix}}$

$={\begin{bmatrix}-2&-4\\9&4\end{bmatrix}}$

Practice Problems

(1) ${\begin{bmatrix}1&0\\2&2\end{bmatrix}}{\begin{bmatrix}4\\2\end{bmatrix}}=$

(2) ${\begin{bmatrix}1&2\\2&3\end{bmatrix}}{\begin{bmatrix}2&3\\1&4\end{bmatrix}}=$

Dot product

A row vector is a 1 × m matrix, while a column vector is a m × 1 matrix.

Suppose A is row vector and B is column vector, then the dot product is defined as follows;

$A\cdot B=|A||B|cos\theta$

or

$\mathbf {A} \cdot \mathbf {B} ={\begin{pmatrix}a_{1}&a_{2}&\cdots &a_{n}\end{pmatrix}}{\begin{pmatrix}b_{1}\\b_{2}\\\vdots \\b_{n}\end{pmatrix}}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}=\sum _{i=1}^{n}a_{i}b_{i}$

Suppose $\mathbf {A} ={\begin{pmatrix}a_{1}&a_{2}&a_{3}\end{pmatrix}}$ and $\mathbf {B} ={\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}}$ The dot product is

$\mathbf {A} \cdot \mathbf {B} ={\begin{pmatrix}a_{1}&a_{2}&a_{3}\end{pmatrix}}{\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$

Example

Suppose $\mathbf {A} ={\begin{pmatrix}2\\1\\3\end{pmatrix}}$ and $\mathbf {B} ={\begin{pmatrix}7\\5\\4\end{pmatrix}}$

$\mathbf {A} \cdot \mathbf {B} ={\begin{pmatrix}2&1&3\end{pmatrix}}{\begin{pmatrix}7\\5\\4\end{pmatrix}}$

$=2\cdot 7+1\cdot 5+3\cdot 4$
$=14+5+12$

$=31$

Practice problems

(1) $\mathbf {A} ={\begin{pmatrix}3\\2\\5\end{pmatrix}}$ and $\mathbf {B} ={\begin{pmatrix}1\\4\\3\end{pmatrix}}$

$\mathbf {A} \cdot \mathbf {B} =$

(2) $\mathbf {A} ={\begin{pmatrix}1\\0\\3\end{pmatrix}}$ and $\mathbf {B} ={\begin{pmatrix}6\\9\\2\end{pmatrix}}$

$\mathbf {A} \cdot \mathbf {B} =$

Cross product

Cross product is defined as follows:

$A\times B=|A||B|sin\theta$

Or, using detriment,

$\mathbf {A\times B} ={\begin{vmatrix}e_{x}&e_{y}&e_{z}\\a_{x}&a_{y}&a_{z}\\b_{x}&b_{y}&b_{z}\\\end{vmatrix}}=(a_{y}b_{z}-a_{z}b_{y},a_{z}b_{x}-a_{x}b_{z},a_{x}b_{y}-a_{y}b_{x})$

where $e$ is unit vector.

Lagrange Equations

Lagrange Equation

${\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)-{\frac {\partial L}{\partial x}}=0$

where ${\dot {x}}={\frac {dx}{dt}}$
The equation above is called Lagrange Equation.

Let the kinetic energy of the point mass be $T$ and the potential energy be $U$.
$T-U$ is called Lagrangian. Then the kinetic energy is expressed by

$T={\frac {1}{2}}m{\dot {x}}^{2}+{\frac {1}{2}}m{\dot {y}}^{2}$

$={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2})$

Thus

$T=T({\dot {x}},{\dot {y}})$

$U=U(x,y)$

Hence the Lagrangian $L$ is

$L=T-U$

$=T({\dot {x}},{\dot {y}})-U(x,y)$

$={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2})-U(x,y)$

Therefore $T$ relies on only ${\dot {x}}$ and ${\dot {y}}$. $U$ relies on only $x$ and $y$. Thus

${\frac {\partial L}{\partial {\dot {x}}}}={\frac {\partial T}{\partial {\dot {x}}}}=m{\dot {x}}$

${\frac {\partial L}{\partial {\dot {y}}}}={\frac {\partial T}{\partial {\dot {y}}}}=m{\dot {y}}$

In the same way, we have

${\frac {\partial L}{\partial x}}=-{\frac {\partial U}{\partial x}}$

${\frac {\partial L}{\partial y}}=-{\frac {\partial U}{\partial y}}$

The State Equation

Ideal gas law

As an experimental rule, about gas, the equation below is found.

$PV=nRT$...(1)

where

$\ p$ = Pressure (absolute)

$\ V$ = Volume

$\ n$ = Number of moles of a substance

$\ T$ = Absolute temperature

$\ R$ = Gas constant

$R\simeq 8.31J(Kmol)^{-1}$

The gas which strictly follows the law (1) is called ideal gas. (1) is called ideal gas law. Ideal gas law is the state equation of gas. In high school, the ideal gas constant below might be used:

$R=0.082LatmK^{-1}mol^{-1}$

But, in university, the gas constant below is often used:

$R=8.31JK^{-1}mol^{-1}$