imported>Chunbum Park |
imported>John Stephenson |
| (200 intermediate revisions by 8 users not shown) |
| Line 1: |
Line 1: |
| The '''[[Dirac delta function]]''' is a function introduced in 1930 by Paul Adrien Maurice Dirac in his seminal book on quantum mechanics. A physical model that visualizes a delta function is a mass distribution of finite total mass ''M''—the integral over the mass distribution. When the distribution becomes smaller and smaller, while ''M'' is constant, the mass distribution shrinks to a ''point mass'', which by definition has zero extent and yet has a finite-valued integral equal to total mass ''M''. In the limit of a point mass the distribution becomes a Dirac delta function.
| | {{:{{FeaturedArticleTitle}}}} |
| | | <small> |
| Heuristically, the Dirac delta function can be seen as an extension of the Kronecker delta from integral indices (elements of <font style="vertical-align: 13%"> <math>\mathbb{Z}</math></font>) to real indices (elements of <font style="vertical-align: 13%"><math>\mathbb{R}</math></font>). Note that the Kronecker delta acts as a "filter" in a summation:
| | ==Footnotes== |
| :<math> | | {{reflist|2}} |
| \sum_{i=m}^n \; f_i\; \delta_{ia} =
| | </small> |
| \begin{cases}
| |
| f_a & \quad\hbox{if}\quad a\in[m,n] \sub\mathbb{Z} \\
| |
| 0 & \quad \hbox{if}\quad a \notin [m,n].
| |
| \end{cases}
| |
| </math> | |
| | |
| In analogy, the Dirac delta function δ(''x''−''a'') is defined by (replace ''i'' by ''x'' and the summation over ''i'' by an integration over ''x''),
| |
| :<math>
| |
| \int_{a_0}^{a_1} f(x) \delta(x-a) \mathrm{d}x =
| |
| \begin{cases}
| |
| f(a) & \quad\hbox{if}\quad a\in[a_0,a_1] \sub\mathbb{R}, \\
| |
| 0 & \quad \hbox{if}\quad a \notin [a_0,a_1].
| |
| \end{cases}
| |
| </math>
| |
| | |
| The Dirac delta function is ''not'' an ordinary well-behaved map <font style="vertical-align: 12%"><math>\mathbb{R} \rightarrow \mathbb{R}</math></font>, but a distribution, also known as an ''improper'' or ''generalized function''. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand ("under the integral"). Mathematicians say that the delta function is a linear functional on a space of test functions.
| |
| | |
| ==Properties==
| |
| Most commonly one takes the lower and the upper bound in the definition of the delta function equal to <math>-\infty</math> and <math> \infty</math>, respectively. From here on this will be done.
| |
| :<math>
| |
| \begin{align}
| |
| \int_{-\infty}^{\infty} \delta(x)\mathrm{d}x &= 1, \\
| |
| \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx} \mathrm{d}k &= \delta(x) \\
| |
| \delta(x-a) &= \delta(a-x), \\
| |
| (x-a)\delta(x-a) &= 0, \\
| |
| \delta(ax) &= |a|^{-1} \delta(x) \quad (a \ne 0), \\
| |
| f(x) \delta(x-a) &= f(a) \delta(x-a), \\
| |
| \int_{-\infty}^{\infty} \delta(x-y)\delta(y-a)\mathrm{d}y &= \delta(x-a)
| |
| \end{align}
| |
| </math> | |
| The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus. The delta function as a Fourier transform of the unit function ''f''(''x'') = 1 (the second property) will be proved below.
| |
| The last property is the analogy of the multiplication of two identity matrices,
| |
| :<math>
| |
| \sum_{j=1}^n \;\delta_{ij}\;\delta_{jk} = \delta_{ik}, \quad i,k=1,\ldots, n.
| |
| </math>
| |
| ''[[Dirac delta function|.... (read more)]]''
| |
Latest revision as of 10:19, 11 September 2020
The Irvin pin. The eyes have always been red, but there are
urban legends about the meanings of other colors.
A pin from another company, possibly Switlik or Standard Parachute. This style is common in catalogs and auctions of military memorabilia.
The Caterpillar Club is an informal association of people who have successfully used a parachute to bail out of a disabled aircraft. After authentication by the parachute maker, applicants receive a membership certificate and a distinctive lapel pin.
History
Before April 28, 1919 there was no way for a pilot to jump out of a plane and then to deploy a parachute. Parachutes were stored in a canister attached to the aircraft, and if the plane was spinning, the parachute could not deploy. Film industry stuntman Leslie Irvin developed a parachute that the pilot could deploy at will from a back pack using a ripcord. He joined the Army Air Corps parachute research team, and in April 1919 he successfully tested his design, though he broke his ankle during the test. Irvin was the first person to make a premeditated free fall jump from an airplane. He went on to form the Irving Airchute Company, which became a large supplier of parachutes. (A clerical error resulted in the addition of the "g" to Irvin and this was left in place until 1970, when the company was unified under the title Irvin Industries Incorporated.) The Irvin brand is now a part of Airborne Systems, a company with operations in Canada, the U.S. and the U.K.[1].
An early brochure [2] of the Irvin Parachute Company credits William O'Connor 24 August 1920 at McCook Field near Dayton, Ohio as the first person to be saved by an Irvin parachute, but this feat was unrecognised. On 20 October 1922 Lieutenant Harold R. Harris, chief of the McCook Field Flying Station, jumped from a disabled Loening W-2A monoplane fighter. Shortly after, two reporters from the Dayton Herald, realising that there would be more jumps in future, suggested that a club should be formed. 'Caterpillar Club' was suggested because the parachute canopy was made of silk, and because caterpillars have to climb out of their cocoons and fly away. Harris became the first member, and from that time forward any person who jumped from a disabled aircraft with a parachute became a member of the Caterpillar Club. Other famous members include General James Doolittle, Charles Lindbergh and (retired) astronaut John Glenn.
In 1922 Leslie Irvin agreed to give a gold pin to every person whose life was saved by one of his parachutes. By 1945 the number of members with the Irvin pins had grown to over 34,000. In addition to the Irvin Air Chute Company and its successors, other parachute manufacturers have also issued caterpillar pins for successful jumps. Irvin/Irving's successor, Airborne Systems Canada, still provides pins to people who made their jump long ago and are just now applying for membership. Another of these is Switlik Parachute Company, which though it no longer makes parachutes, still issues pins.
