text:
Lava Beds National Monument

Lava Beds National Monument is located in northeastern California, in Siskiyou and Modoc counties. The monument lies on the northeastern flank of Medicine Lake Volcano, which is the largest volcano by area in the Cascade Range.

The region in and around Lava Beds National Monument lies at the junction of the Sierra-Klamath, Cascade, and Great Basin physiographic provinces. The monument was established as a national monument on November 21, 1925, and includes more than 46000 acre.

Lava Beds National Monument has numerous lava tubes, with 27 having marked entrances and developed trails for public access and exploration. The monument also offers trails through the high Great Basin xeric shrubland desert landscape and the volcanic field. In 1872 and 1873, the area was the site of the Modoc War, involving a band led by Kintpuash (also known as Captain Jack). The area of Captain Jack's Stronghold was named in his honor.

Geologic formations
Lava Beds National Monument is geologically significant because of its wide variety of volcanic formations, including lava tubes, fumaroles, cinder cones, spatter cones, pit craters, hornitos, maars, lava flows, and volcanic fields.

Volcanic eruptions on the Medicine Lake shield volcano have created a rugged landscape punctuated by these many landforms of volcanism.

Cones
Cinder cones are formed when magma is under great pressure. It is released in a fountain of lava, blown into the air from a central vent. The lava cools as it falls, forming cinders that pile up around the vent. When the pressure has been relieved, the rest of the lava flows from the base of the cone. Cinder cones are typically monogenetic.

The cinder cones of Hippo Butte, Three Sisters, Juniper Butte, and Crescent Butte are all older than the Mammoth and Modoc Crater flows, more than 30,000–40,000 years old. Eagle Nest Butte and Bearpaw Butte are 114,000 years old. Schonchin Butte cinder cone and the andesitic flow from its base were formed around 62,000 years ago. The flow that formed Valentine Cave erupted 10,850 years ago. An eruption that formed The Castles is younger than the Mammoth Crater flows. Even younger were eruptions from Fleener Chimneys, such as the Devil's Homestead flow, 10,500 years ago, and Black Crater 3,025 years ago. About 1,110 years ago, plus or minus 60 years, the Callahan flow was produced by an eruption from Cinder Butte. Though Cinder Butte is just outside the boundary of the monument, the Callahan flow is in Lava Beds and is the youngest flow in the monument.

Spatter cones are built out of thicker lava. The lava is thrown out of the vent and builds, layer by layer, a chimney surrounding the vent. Fleener Chimneys and Black Crater are examples of spatter cones.

Lava flows
Roughly ninety percent of the lava in the Lava Beds Monument is basaltic. There are primarily two kinds of basaltic lava flows: pahoehoe and ʻaʻā. Pahoehoe is smooth, often ropy and is the most common type of lava in Lava Beds. ʻAʻā is formed when pahoehoe cools and loses some of its gases. ʻAʻā is rough, sharp, and jagged; an excellent example is the Devil's Homestead lava flow, which originated at Fleener Chimneys. Most of the rest of the lava in the monument is andesitic. Pumice, a type of rhyolitic lava, also is found covering the monument; this rained down around 900 years ago during the eruption of Glass Mountain.

The flows from Mammoth and Modoc Craters comprise about two-thirds of the lava in the monument. Over 30 separate lava flows located in the park range in age from 2,000,000 years BP to 1,110 years BP. Some of the major lava flows within Lava Beds National Monument include the Callahan Flow; Schonchin Flow; Mammoth Crater Flow; Modoc Crater Flow; and Devil's Homestead Flow.

Lava fields
Gillem Bluff, a fault scarp, was created as the region stretched and a block of earth dropped down along this fault (see Basin and Range Province). The tuff layer on top of Gillem Bluff is 2,000,000 years old, indicating the rock layers beneath are even older. The oldest lava flow from the Medicine Lake Volcano within the monument is the Basalt of Hovey Point, near Captain Jack's Stronghold, which is 450,000 years old. Petroglyph Point was created about 275,000 years ago when cinders erupted through the shallow water of Tule Lake; violent explosions of ash and steam formed layers upon layers of tuff.

The caldera is thought to have formed by subsidence, during which basalt and andesite were erupted up on the slopes.

Lava tubes
Lava flows dated to about 30,000–40,000 years ago formed most of the lava tubes in the monument. As the hot basaltic lava flowed downhill, the top cooled and crusted over, insulating the rest of the lava and forming lava tubes. Lavacicles on the ceiling of a lava tube were produced as the level of lava in the tube retreated and the viscous lava on the ceiling dripped as it cooled.

Dripstone was created when lava splashed on the inside walls of the tubes. The leaching of minerals from pumice gravel, soils, and overlying rock provides for deposition of secondary speleothems in lava tubes.

Lava Beds National Monument has the largest concentration of lava tubes in North America. One has electrical lighting; the others are illuminated by ceiling collapse portals or require flashlights, available on loan.

Notable examples in the park include Catacombs Cave, Merrill Cave, Mushpot Cave, and Valentine Cave.

Recent activity
A series of small earthquakes in late 1988 has been attributed to subsidence in the caldera. North-northeast trending ground cracks, as well as north-northeast trending vent series show relationships between tectonism and volcanism. One notable ground crack, the Big Crack, extends along the northeastern boundary of the monument.

Climate
The high elevation, semi-arid desert environment of Lava Beds National Monument receives an average of 14.22 in of annual precipitation, including 43.2 in of snowfall. The climate is characterized by warm, dry summers and cold, moderately snowy winters. The monthly daily average temperature ranges from 31.8 °F in December to 68.92 °F in July; there are an average of 23.8 days with 90 °F+ highs annually, and 8.4 days where the high does not rise above freezing. The average window for freezing temperatures is September 20 thru June 6.

Flora
The lava tube collapse systems and lava outcrops support a great diversity of plant life, from an impressive variety of lichens and mosses to plants such as desert sweet (Chamaebatiaria millefolium), the aromatic purple desert sage (Salvia dorrii carnosa) and yellow blazing star (Mentzelia laevicaulis). An impressive variety of fern species are present in cave entrances including the spreading wood fern (Dryopteris expansa) and the western swordfern (Polystichum munitum). These species are well outside of their normal range, which is 90 - 125 mi west on the northern California coastline.

Fauna
Despite harsh, semi-arid conditions, native wildlife has adapted to the environmental constraints present in the region. There are no terrestrial water resources in Lava Beds National Monument. Some animals obtain water from caves, while others fly about twenty km (12 miles) north to Tule Lake. Federal and state animal species of special concern in the Monument include the Cooper's hawk (Accipiter cooperii), fringed myotis (Myotis thysanodes), long-eared myotis (Myotis evotis), long-legged myotis (Myotis volans), pallid bat (Antrozous pallidus), silver-haired bat (Lasionycteris noctivigans), Townsends big-eared bat (Corynorhinus townsendii), western small-footed myotis (Myotis ciliolabrum), and American badger (Taxidea taxus).

Because of a lack of surface water, amphibian presence in the monument is limited. The most common species found in the monument is the Pacific tree frog (Pseudacris regilla). This species is also found in the biologically rich cave entrances in the monument. Reptile species found in the monument include the northern sagebrush lizard (Sceloporus graciosus graciosus), Great Basin fence lizard (Sceloporus occidentalis biseriatus), western skink (Eumeces skiltonianus skiltonianus), Rocky Mountain rubber boa (Charina bottae utahensis), gopher snake (Pituophis melanoleucus), desert night snake (Hypsiglena torquata deserticola), and western rattlesnake (Crotalus viridis).

Key animal species by habitat:
 * Bunch grass-sagebrush grasslands: mule deer, gopher snake, coyote, prairie falcon, western meadowlark, Anderson's larkspur, yellow-bellied marmot, kangaroo rat
 * Ponderosa-juniper brushlands: pronghorn, bobcat, scrub jay, jackrabbit, mountain bluebird,
 * Pine forest: mountain lion, bald eagle (winter resident), golden-mantled ground squirrel
 * Rocky lava flows: western fence lizard, western rattlesnake
 * Cave mouths and interior passages: violet-green swallow, Brazilian free-tailed bat, Townsend's big-eared bat, Pacific tree frog, pika, bushy-tailed woodrat
 * Lake shore areas: skunk, raccoon, great horned owl, short-eared owl.

Archeological site
Lava Beds National Monument includes Petroglyph Point, one of the largest panels of Native American rock art in the United States. The region was historically occupied by the Modoc people. The Lava Beds National Monument Archeological District was listed on the National Register of Historic Places in March 1991.

Modoc War

During the Modoc War of 1872–1873, warriors of a band led by Kintpuash (Captain Jack) used the lava beds as a defensive stronghold to resist being captured and returned to the Klamath Reservation in Oregon, to which they had been removed, as European Americans wanted their lands. The Modoc took refuge in a natural lava fortress that was later named Captain Jack's Stronghold. From this defensive base, a group of 53 fighting men and their families held off US Army forces, amounting to ten times the Modocs' population, for five months.

In April 1873, at a peace commission meeting, Captain Jack killed General Edward Canby while associates killed Reverend Eleazer Smith and wounded two other commissioners. Canby was the only casualty of the Indian Wars who had the rank of general at the time of his death. The Modoc mistakenly thought the Americans would leave if their leaders were killed. Instead Army reinforcements were brought in and the Modoc were eventually forced to surrender. Kintpuash and his associates were convicted of war crimes for the murders, and executed by hanging at Fort Klamath.

Establishment of the monument
An early steward for the land was J. D. Howard, a miller residing in Klamath Falls. He explored the area intimately beginning in 1917, naming geographic features and making caves accessible. With the area growing attention, he noticed visitors beginning to damage the cave features.

Howard's perceived need to preserve the lava tubes and the area's other geological features was a major reason why he vigorously pushed for a federally protected status. He would photograph interiors of each cave as part of an effort to persuade people to protect the area. In a letter written in 1923, Howard urged United States Forest Service officials, state legislators, and community leaders to declare the caves a park; "This is the last of Park sites in the States, and is far Superior to all the others as it embraces enough phenomena to keep one busy for at least Three Month Sight Seeing with Cap Jack’s Stronghold to study on leisure hours."

Howard's campaign was successful, as Lava Beds National Monument was signed into existence by President Calvin Coolidge on November 21, 1925. It was created under the auspices of the 1906 Antiquities Act.

Later expansion
The only expansion in the modern era occurred in 2011 when the monument expanded by roughly 132 acre. An administrative transfer brought two parcels of land which were previously managed by the Bureau of Reclamation and the Bureau of Land Management into National Park Service management. These two parcels adjoin the Petroglyph Point Unit of the monument.

Lava Beds Wilderness
The Lava Beds National Wilderness is a 28460 acre wilderness area within the Lava Beds National Monument. It was designated by the US Congress on October 13, 1972, with passage of Public Law 92-493. The wilderness protects more than half of the national monument in two separate eastern and western units. The larger eastern unit contains the extensive Schonchin lava flow on the east side of the monument. The western unit covers the monument's area within the transition zone of the Cascade Range's montane southern end and the arid Modoc Plateau ecosystems.

The different soil types creates plant community diversity in the wilderness area, providing various habitats for a wide range of wildlife. The numerous coyotes and foxes, as well as raptors, feed on rodents such as the jackrabbit and kangaroo rat. The kangaroo rat is especially adapted to waterless environments because it does not need to drink. It obtains water as a byproduct of the oxidation of nutrients in the seeds it eats.

Many raptors are seen in the wilderness area, with 24 species of hawks identified. The monument is located on the Pacific Flyway, and the bald eagle winters in the northern portion of the wilderness area.

The National Park Service manages the Lava Beds wilderness area and has several restrictions in place. Camping is prohibited near cave entrances or trails. Open campfires may be prohibited during very hot and dry weather.

Lava Beds trails
The Lava Beds National Monument has 13 hiking trails, all of which cross or enter the backcountry. The most popular trails are short, but lead to a number of historic sites as to several geological areas within the Lava Beds Wilderness. The long trails are mostly in designated wilderness areas.

These trails are primarily associated with park attractions, especially the Lava Beds geological wilderness, and are sometimes out-and-back day hikes, and in many cases are improved boardwalks and supported with interpretive signs and exhibits.

title: Lava Beds National Monument


text:
Talk:Game.com

Untitled
The links "handheld" and "console" should probably target the dedicated page hand held console.
 * Malcohol 09:53 9 Jul 2003 (UTC)

First Internet Console
Sega Saturn's Netlink was released in 1996, a year before Game.com launched, and provided users the ability to surf the web, among other things. I didn't change the information here, but I believe what is listed is incorrect. Thanks.

 * No, it clearly and correctly says it's the first console to "include" it. Please sign your post. — Smuckola(talk) 03:18, 29 April 2021 (UTC)

Two cartridge slots
I am curious about the statement "although later models reverted to the industry-standard single slot." I bought my game.com after manufacturing was halted and when the retailers were heavily discounting them to clear out the inventory, but mine has two cartridge slots. Were many of the single slot versions made? -Armaced 01:05, 18 January 2006 (UTC)

Your game.com might have been on the shelf before the release of the game.com PocketPro.
 * The big one showed in the pic has two slots, a smaller version, then a smaller version with a backlight were released, both with only one cart slot. --Thaddius 13:04, 29 June 2006 (UTC)

i bought mine a long time ago and it broke 3 weeks after i got it. it was not in the box though it was in plastic and did not come with a game so i bought duke nuke-um 3d and the arcade one.after it broke i returned to the store and ask them if they had a another one and they laught at me. over all it was a crap hand held....

In my experience with the two slot system, some permutations of cartridges just didn't work! Either the game startup animation would freeze, or one/both carts wouldn't be detected. 68.0.226.163 02:00, 23 January 2007 (UTC)
 * That's probably due to a poor connection 'tween the cart and the system. It's possible that the contacts on those games you tried are dirty. It's also possible that the system sucks (the latter is more likely). --Thaddius (talk) 13:16, 8 September 2008 (UTC)

Slight edit
I modified one sentence in the game.com article.

Original version:

"When game.com was discontinued, no handheld games console would be made with a touchscreen until the coming of the Nintendo DS."

My new version:

"After the game.com was discontinued, the next handheld game console to feature a touch screen was the Nintendo DS, which debuted in 2004."

The new sentence has a better structure and is more specific.

-R

 * That part seems extremely irrelevant, and as it's written currently it creates a false image of the DS (it suggests the DS is a rip-off of the Game.com when in fact the DS' second cartridge slot is for GBA games and its internet abilities are the same as other modern consoles, very different from game.com's). I'm getting rid of it. -Unknownwarrior33 01:07, 14 September 2006 (UTC)

 * R doesn't even mention the second cartridge slot, what are you worried about? —The preceding unsigned comment was added by 212.146.47.250 (talk) 11:43, 13 May 2007 (UTC).

Maybe the European (England) release should be mentioned, the game.com was sold with 'Lights Out' and 'Batman and Robin' included, as seen on the packing box: (http://videogamecollectors.com/gallery/album345/tiger_game_com_1). Also, if you purchased the game.com upon release in the UK, you received a letter from Tiger with an apology stating that 'Batman and Robin' is not available yet, and we enclosed a free copy of 'Wheel of Fortune' and a SASE to send to us for sending you B&R in return when it becomes available. Basically, you got three free games with game.com in UK: http://videogamecollectors.com/gallery/album345/tiger_game_com_games_8

Cancelled games

 * A Bug's Life
 * Castlevania: Symphony of the Night
 * Command and Conquer: Red Alert
 * Deer Hunter
 * Furbyland
 * GigaPets Deluxe
 * Godzilla
 * Holyfield Boxing
 * Madden Football '99
 * Metal Gear Solid
 * Mulligan's Golf Challenge
 * Mutoids
 * Name That Tune
 * NASCAR Racing
 * NBA Hangtime
 * NBA Live '99
 * Pocket Fisherman
 * The Price is Right
 * Rugrats
 * Small Soldiers
 * Turok: Dinosaur Hunter
 * WCW Whiplash
 * The X-Files

Can anyone find any reliable sources stating that these games were planned to be released? All I can find are sketchy fansites and usenet posts, neither of which really pass WP:RS muster. - A Man In Bl♟ck (conspire | past ops) 19:35, 14 August 2006 (UTC)

A game.com enthusiast with Usenet access created the newsgroup alt.games.video.tiger.game-com (Google Groups link below) shortly after the handheld's release in 1997. This group served as a focus point for game.com owners, with often-heated discussion about the handheld's future and merits relative to other systems; as well as reviews of existing games and speculation about future releases. Tiger representatives sometimes posted using the now-defunct America Online account "TigerGcom." At one point, a gameplay video of the never-released Metal Gear Solid was distributed among group members. Most of the information about unreleased games herein was gleaned from postings to alt.games.video.tiger.game-com.

Likewise this; do we have any independent verification that any of this is true, beyond the usenet posts themselves? - A Man In Bl♟ck (conspire | past ops) 19:36, 14 August 2006 (UTC)

 * I've marked the two sections you've mentioned as OR until I can get around to sourcing them. Most of the unreleased games were announced via the box art/packaging on games that were released for game.com. TheNewMinistry 22:43, 14 August 2006 (UTC)
 * Please don't put them back into the article until they're sourced to a reliable source. They've been up for years without a reference, and, frankly, it's an embarassment. - A Man In Bl♟ck (conspire | past ops) 22:48, 14 August 2006 (UTC)
 * More specifically, most of these games were detailed in a small catalog that came with the last wave of releases for the console. However, it doesn't seem prudent to me to cite something that someone would have to scour eBay for, and the catalog probably can't be included in the Wikimedia Commons. So it may be best to wait until it is hosted offsite someplace and can be linked to. istewart 09:28, 13 September 2006 (UTC)
 * I would regard such a catalog as a perfectly valid resource, regardless of whether it is hosted on-line.--Malcohol 11:59, 13 April 2007 (UTC)

IGN and game.com version
http://search.ign.com/products?qtype=0&query=Metal+Gear+Solid+game.com&objtName=all&sort=date&so=exact&ns=true&genNav=true&x=0&y=0 75.4.200.230 22:52, 9 February 2007 (UTC)

Game commies
It could be worth mentioning that the hacker group "game commies" gets their "commies" name from the first post on game.com's newsgroup. If you read the messsage somebody replies and makes a joke about people who like the "com" would be called "Commies" and then were like uhhh maybe not. It's post 2422 i believe if you are wondering. —Preceding unsigned comment added by 24.129.235.101 (talk) 23:46, 22 August 2009 (UTC)

Tiger-Game-Com-FL.jpg}} to appear as POTD soon
Hello! This is a note to let the editors of this article know that File: will be appearing as picture of the day on September 6, 2018. You can view and edit the POTD blurb at Template:POTD/2018-09-06. If this article needs any attention or maintenance, it would be preferable if that could be done before its appearance on the Main Page so Wikipedia doesn't look bad. :) Thanks! MER-C 10:34, 2 September 2018 (UTC)

A Commons file used on this page or its Wikidata item has been nominated for deletion
The following Wikimedia Commons file used on this page or its Wikidata item has been nominated for deletion: Participate in the deletion discussion at the. —Community Tech bot (talk) 07:31, 30 June 2021 (UTC)
 * GameComLightsOutCart.jpg

First touch screen console
The Bandai Denshi Manga Juku and Philips In2it both predate this console, and both contain touchscreens. This conflicts with the claim that "The Game.com was the first video game console to feature a touchscreen" Porotoman99 (talk) 00:34, 30 October 2023 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Game.com


text:
Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).

This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.

Definition
Given a nonnegative integer m, an order-$$m$$ linear differential operator is a map $$P$$ from a function space $$\mathcal{F}_1$$ to another function space $$\mathcal{F}_2$$ that can be written as:

$$P = \sum_{|\alpha|\le m}a_\alpha(x) D^\alpha\ ,$$ where $$\alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n)$$ is a multi-index of non-negative integers, $$|\alpha| = \alpha_1 + \alpha_2 + \cdots + \alpha_n$$, and for each $$\alpha$$, $$a_\alpha(x)$$ is a function on some open domain in n-dimensional space. The operator $$D^\alpha$$ is interpreted as

$$D^\alpha = \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}}$$

Thus for a function $$f \in \mathcal{F}_1$$:

$$P f = \sum_{|\alpha|\le m}a_\alpha(x) \frac{\partial^{|\alpha|}f}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}}$$

The notation $$D^{\alpha}$$ is justified (i.e., independent of order of differentiation) because of the symmetry of second derivatives.

The polynomial p obtained by replacing D by variables $$\xi$$ in P is called the total symbol of P; i.e., the total symbol of P above is: $$p(x, \xi) = \sum_{|\alpha|\le m}a_\alpha(x) \xi^\alpha$$ where $$\xi^\alpha = \xi_1^{\alpha_1} \cdots \xi_n^{\alpha_n}.$$ The highest homogeneous component of the symbol, namely,
 * $$\sigma(x, \xi) = \sum_{|\alpha|= m}a_\alpha(x) \xi^\alpha$$

is called the principal symbol of P. While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle).

More generally, let E and F be vector bundles over a manifold X. Then the linear operator

 * $$ P: C^\infty(E) \to C^\infty(F) $$

is a differential operator of order $$ k $$ if, in local coordinates on X, we have

 * $$ Pu(x) = \sum_{|\alpha| = k} P^\alpha(x) \frac {\partial^\alpha u} {\partial x^{\alpha}} + \text{lower-order terms}$$

where, for each multi-index α, $$ P^\alpha(x):E \to F$$ is a bundle map, symmetric on the indices α.

The kth order coefficients of P transform as a symmetric tensor

 * $$ \sigma_P: S^k (T^*X) \otimes E \to F $$

whose domain is the tensor product of the kth symmetric power of the cotangent bundle of X with E, and whose codomain is F. This symmetric tensor is known as the principal symbol (or just the symbol) of P.

The coordinate system xi permits a local trivialization of the cotangent bundle by the coordinate differentials dxi, which determine fiber coordinates ξi. In terms of a basis of frames eμ, fν of E and F, respectively, the differential operator P decomposes into components

 * $$(Pu)_\nu = \sum_\mu P_{\nu\mu}u_\mu$$

on each section u of E. Here Pνμ is the scalar differential operator defined by

 * $$P_{\nu\mu} = \sum_{\alpha} P_{\nu\mu}^\alpha\frac{\partial}{\partial x^\alpha}.$$

With this trivialization, the principal symbol can now be written

 * $$(\sigma_P(\xi)u)_\nu = \sum_{|\alpha|=k} \sum_{\mu}P_{\nu\mu}^\alpha(x)\xi_\alpha u_\mu.$$

In the cotangent space over a fixed point x of X, the symbol $$ \sigma_P $$ defines a homogeneous polynomial of degree k in $$ T^*_x X $$ with values in $$ \operatorname{Hom}(E_x, F_x) $$.

Fourier interpretation
A differential operator P and its symbol appear naturally in connection with the Fourier transform as follows. Let ƒ be a Schwartz function. Then by the inverse Fourier transform,

 * $$Pf(x) = \frac{1}{(2\pi)^{\frac{d}{2}}} \int\limits_{\mathbf{R}^d} e^{ ix\cdot\xi} p(x,i\xi)\hat{f}(\xi)\, d\xi.$$

This exhibits P as a Fourier multiplier. A more general class of functions p(x,ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises the pseudo-differential operators.

Examples

 * The differential operator $$ P $$ is elliptic if its symbol is invertible; that is for each nonzero $$ \theta \in T^*X $$ the bundle map $$ \sigma_P (\theta, \dots, \theta)$$ is invertible. On a compact manifold, it follows from the elliptic theory that P is a Fredholm operator: it has finite-dimensional kernel and cokernel.
 * In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation.
 * In applications to the physical sciences, operators such as the Laplace operator play a major role in setting up and solving partial differential equations.
 * In differential topology, the exterior derivative and Lie derivative operators have intrinsic meaning.
 * In abstract algebra, the concept of a derivation allows for generalizations of differential operators, which do not require the use of calculus. Frequently such generalizations are employed in algebraic geometry and commutative algebra. See also Jet (mathematics).
 * In the development of holomorphic functions of a complex variable z = x + i y, sometimes a complex function is considered to be a function of two real variables x and y. Use is made of the Wirtinger derivatives, which are partial differential operators: $$ \frac{\partial}{\partial z} = \frac{1}{2} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) \ ,\quad \frac{\partial}{\partial\bar{z}}= \frac{1}{2} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) \ .$$ This approach is also used to study functions of several complex variables and functions of a motor variable.
 * The differential operator del, also called nabla, is an important vector differential operator. It appears frequently in physics in places like the differential form of Maxwell's equations. In three-dimensional Cartesian coordinates, del is defined as
 * $$\nabla = \mathbf{\hat{x}} {\partial \over \partial x} + \mathbf{\hat{y}} {\partial \over \partial y} + \mathbf{\hat{z}} {\partial \over \partial z}.$$
 * Del defines the gradient, and is used to calculate the curl, divergence, and Laplacian of various objects.

 * A chiral differential operator. For now, see

History
The conceptual step of writing a differential operator as something free-standing is attributed to Louis François Antoine Arbogast in 1800.

Notations
The most common differential operator is the action of taking the derivative. Common notations for taking the first derivative with respect to a variable x include:

 * $${d \over dx}$$, $$D$$, $$D_x,$$ and $$\partial_x$$.

When taking higher, nth order derivatives, the operator may be written:

 * $${d^n \over dx^n}$$, $$D^n$$, $$D^n_x$$, or $$\partial_x^n$$.

The derivative of a function f of an argument x is sometimes given as either of the following:

 * $$[f(x)]'$$
 * $$f'(x).$$

The D notation's use and creation is credited to Oliver Heaviside, who considered differential operators of the form

 * $$\sum_{k=0}^n c_k D^k$$

in his study of differential equations.

One of the most frequently seen differential operators is the Laplacian operator, defined by

 * $$\Delta = \nabla^2 = \sum_{k=1}^n \frac{\partial^2}{\partial x_k^2}.$$

Another differential operator is the Θ operator, or theta operator, defined by

 * $$\Theta = z {d \over dz}.$$

This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z: $$\Theta (z^k) = k z^k,\quad k=0,1,2,\dots $$

In n variables the homogeneity operator is given by $$\Theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}.$$

As in one variable, the eigenspaces of Θ are the spaces of homogeneous functions. (Euler's homogeneous function theorem)

In writing, following common mathematical convention, the argument of a differential operator is usually placed on the right side of the operator itself. Sometimes an alternative notation is used: The result of applying the operator to the function on the left side of the operator and on the right side of the operator, and the difference obtained when applying the differential operator to the functions on both sides, are denoted by arrows as follows:
 * $$f \overleftarrow{\partial_x} g = g \cdot \partial_x f$$
 * $$f \overrightarrow{\partial_x} g = f \cdot \partial_x g$$
 * $$f \overleftrightarrow{\partial_x} g = f \cdot \partial_x g - g \cdot \partial_x f.$$

Such a bidirectional-arrow notation is frequently used for describing the probability current of quantum mechanics.

Adjoint of an operator
Given a linear differential operator $$T$$ $$Tu = \sum_{k=0}^n a_k(x) D^k u$$ the adjoint of this operator is defined as the operator $$T^*$$ such that $$\langle Tu,v \rangle = \langle u, T^*v \rangle$$ where the notation $$\langle\cdot,\cdot\rangle$$ is used for the scalar product or inner product. This definition therefore depends on the definition of the scalar product (or inner product).

Formal adjoint in one variable
In the functional space of square-integrable functions on a real interval $(a, b)$, the scalar product is defined by $$\langle f, g \rangle = \int_a^b \overline{f(x)} \,g(x) \,dx, $$

where the line over f(x) denotes the complex conjugate of f(x). If one moreover adds the condition that f or g vanishes as $$x \to a$$ and $$x \to b$$, one can also define the adjoint of T by $$T^*u = \sum_{k=0}^n (-1)^k D^k \left[ \overline{a_k(x)} u \right].$$

This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When $$T^*$$ is defined according to this formula, it is called the formal adjoint of T.

A (formally) self-adjoint operator is an operator equal to its own (formal) adjoint.

Several variables
If Ω is a domain in Rn, and P a differential operator on Ω, then the adjoint of P is defined in L2(Ω) by duality in the analogous manner:

 * $$\langle f, P^* g\rangle_{L^2(\Omega)} = \langle P f, g\rangle_{L^2(\Omega)}$$

for all smooth L2 functions f, g. Since smooth functions are dense in L2, this defines the adjoint on a dense subset of L2: P* is a densely defined operator.

Example
The Sturm–Liouville operator is a well-known example of a formal self-adjoint operator. This second-order linear differential operator L can be written in the form

 * $$Lu = -(pu')'+qu=-(pu+p'u')+qu=-pu-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u.$$

This property can be proven using the formal adjoint definition above.

This operator is central to Sturm–Liouville theory where the eigenfunctions (analogues to eigenvectors) of this operator are considered.

Properties of differential operators
Differentiation is linear, i.e.

 * $$D(f+g) = (Df)+(Dg),$$
 * $$D(af) = a(Df),$$

where f and g are functions, and a is a constant.

Any polynomial in D with function coefficients is also a differential operator. We may also compose differential operators by the rule

 * $$(D_1 \circ D_2)(f) = D_1(D_2(f)).$$

Some care is then required: firstly any function coefficients in the operator D2 must be differentiable as many times as the application of D1 requires. To get a ring of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be commutative: an operator gD isn't the same in general as Dg. For example we have the relation basic in quantum mechanics:
 * $$Dx - xD = 1.$$

The subring of operators that are polynomials in D with constant coefficients is, by contrast, commutative. It can be characterised another way: it consists of the translation-invariant operators.

The differential operators also obey the shift theorem.

Ring of univariate polynomial differential operators
If R is a ring, let $$R\langle D,X \rangle$$ be the non-commutative polynomial ring over R in the variables D and X, and I the two-sided ideal generated by DX − XD − 1. Then the ring of univariate polynomial differential operators over R is the quotient ring $$R\langle D,X\rangle/I$$. This is a non-commutative simple ring. Every element can be written in a unique way as a R-linear combination of monomials of the form $$X^a D^b \text{ mod } I$$. It supports an analogue of Euclidean division of polynomials.

Differential modules over $$R[X]$$ (for the standard derivation) can be identified with modules over $$R\langle D,X\rangle/I$$.

Ring of multivariate polynomial differential operators
If R is a ring, let $$R\langle D_1,\ldots,D_n,X_1,\ldots,X_n\rangle$$ be the non-commutative polynomial ring over R in the variables $$D_1,\ldots,D_n,X_1,\ldots,X_n$$, and I the two-sided ideal generated by the elements
 * $$(D_i X_j-X_j D_i)-\delta_{i,j},\ \ \ D_i D_j -D_j D_i,\ \ \ X_i X_j - X_j X_i$$

for all $$1 \le i,j \le n,$$ where $$\delta$$ is Kronecker delta. Then the ring of multivariate polynomial differential operators over R is the quotient ring $R\langle D_1,\ldots,D_n,X_1,\ldots,X_n\rangle/I$.

This is a non-commutative simple ring. Every element can be written in a unique way as a R-linear combination of monomials of the form $X_1^{a_1} \ldots X_n^{a_n} D_1^{b_1} \ldots D_n^{b_n}$.

Coordinate-independent description
In differential geometry and algebraic geometry it is often convenient to have a coordinate-independent description of differential operators between two vector bundles. Let E and F be two vector bundles over a differentiable manifold M. An R-linear mapping of sections P : Γ(E) → Γ(F) is said to be a kth-order linear differential operator if it factors through the jet bundle Jk(E). In other words, there exists a linear mapping of vector bundles

 * $$i_P: J^k(E) \to F$$

such that

 * $$P = i_P\circ j^k$$

where jk: Γ(E) → Γ(Jk(E)) is the prolongation that associates to any section of E its k-jet.

This just means that for a given section s of E, the value of P(s) at a point x ∈ M is fully determined by the kth-order infinitesimal behavior of s in x. In particular this implies that P(s)(x) is determined by the germ of s in x, which is expressed by saying that differential operators are local. A foundational result is the Peetre theorem showing that the converse is also true: any (linear) local operator is differential.

Relation to commutative algebra
An equivalent, but purely algebraic description of linear differential operators is as follows: an R-linear map P is a kth-order linear differential operator, if for any k + 1 smooth functions $$f_0,\ldots,f_k \in C^\infty(M)$$ we have

 * $$[f_k,[f_{k-1},[\cdots[f_0,P]\cdots]]=0.$$

Here the bracket $$[f,P]:\Gamma(E)\to \Gamma(F)$$ is defined as the commutator

 * $$[f,P](s)=P(f\cdot s)-f\cdot P(s).$$

This characterization of linear differential operators shows that they are particular mappings between modules over a commutative algebra, allowing the concept to be seen as a part of commutative algebra.

A differential operator of infinite order
A differential operator of infinite order is (roughly) a differential operator whose total symbol is a power series instead of a polynomial.

Bidifferential operator
A differential operator acting on two functions $$D(g,f)$$ is called a bidifferential operator. The notion appears, for instance, in an associative algebra structure on a deformation quantization of a Poisson algebra.

Microdifferential operator
A microdifferential operator is a type of operator on an open subset of a cotangent bundle, as opposed to an open subset of a manifold. It is obtained by extending the notion of a differential operator to the cotangent bundle.

title: Differential operator


text:
Cape teal

The Cape teal (Anas capensis) also Cape wigeon or Cape widgeon is a 44–46 cm long dabbling duck of open wetlands in sub-Saharan Africa.

Taxonomy
The Cape teal was formally described in 1789 by the German naturalist Johann Friedrich Gmelin in his revised and expanded edition of Carl Linnaeus's Systema Naturae. He placed it with all the ducks, geese and swans in the genus Anas and coined the binomial name Anas capensis. Gmelin based his description on the "Cape widgeon" that had been described in 1785 by the English ornithologist John Latham in his A General Synopsis of Birds. The naturalist Joseph Banks had provided Latham with a water-colour drawing of the teal by Georg Forster who had accompanied James Cook on his second voyage to the Pacific Ocean. His picture was drawn in 1773 at the Cape of Good Hope. This picture is now the holotype for the species and is held by the Natural History Museum in London. The genus name Anas is the Latin word for a duck. The specific epithet capensis indicates the Cape of Good Hope. The species is monotypic: no subspecies are recognised.

Description
Like many southern ducks, the sexes are similar. It is very pale and mainly grey, with a browner back and pink on the bill (young birds lack the pink). This is a generally quiet species, except during mating displays. The breeding male has a clear whistle, whereas the female has a feeble "quack". The Cape teal cannot be confused with any other duck in its range.

This species is essentially non-migratory, although it moves opportunistically with the rains. It is a thinly distributed but widespread duck, rarely seen in large groups except the moulting flocks, which may number up to 2000.

The Cape teal feeds on aquatic plants and small creatures (invertebrates, crustaceans and amphibians) obtained by dabbling. The nest is on the ground under vegetation and near water.

The Cape teal is one of the species to which the Agreement on the Conservation of African-Eurasian Migratory Waterbirds (AEWA) applies.

title: Cape teal


text:
Tertiary education fees in Australia

Tertiary education fees in Australia are payable for courses at tertiary education institutions. Responsibility for fees in vocational education and training (VET) rests primarily with the state and territory governments, while fees policy in higher education is largely controlled by the Commonwealth Government.

For most domestic students in higher education, the Commonwealth Government provides loans, subsidies, and/or social security welfare payments & benefits to relieve the cost of tertiary education. These benefits are not available to international students. Some domestic students are supported by the government and are required to pay only part of the cost of tuition, called the "student contribution", and the government pays the balance. Some government supported students can defer payment of their contribution as a HECS-HELP loan. Other domestic students are full fee-paying (non-Commonwealth supported) and do not receive direct government contribution to the cost of their education. Some domestic students in full fee courses can obtain a FEE-HELP loan from the Australian government up to a lifetime limit of $150,000 for medicine, dentistry and veterinary science programs and $104,440 for all other programs.

Student fees for vocational education vary between jurisdictions, with some states implementing fee-free courses in some fields and all offering some form of government subsidised training. Funding responsibilities for student fee subsidies are agreed between state and territory governments and the Commonwealth under the National Skills Agreement which commenced at the start of 2024.

Australian citizens (and in some cases overseas professionals completing bridging studies in order to be accredited permanent residents ) are able to obtain loans from the government under the Higher Education Loan Programme (HELP) which replaced the Higher Education Contribution Scheme (HECS). As of April 2016, the amount of money owed to the Australian government under the HECS scheme was AUD$60 billion and is expected to increase to $180 billion by 2026.

HELP is jointly administered by the Australian Department of Education, Skills and Employment and the Australian Taxation Office (ATO). In addition, qualified students may be entitled to Youth Allowance or Austudy Payment to assist them financially while they are studying. These support payments are means and assets tested. Further assistance is available in the form of scholarships. Overseas students are charged fees for the full cost of their education and are ineligible for HELP loans, but may apply for international scholarships.

History
In 1940, the Curtin Labor dramatically increased the number of scholarships to increase the number of university graduates and allowed women avail these scholarships as they were previously exclusive to men. In the 1960s, the Menzies Liberal rapidly established new universities, mostly in outlying suburbs, and offered special research scholarships to encourage students to undertake postgraduate research studies. Many of these universities are members of Innovative Research Universities Australia. In 1967, the government created a category of Commonwealth-funded non-university tertiary institution, called College of Advanced Education (CAE), to provide cheaper & easier access to equivalent of bachelor's degrees.

In 1970s, the Whitlam Labor government abolished university fees to make tertiary education in Australia more accessible to working and middle class Australians. In 1989, the Hawke Labor government began gradually re-introducing fees for university study and setup the Higher Education Contributions Scheme (HECS).

Present status
In 1996, the new Howard Coalition government, introduced tiers in the HECS fee structure (now called HECS-HELP). Fees are charged on the basis of the perceived value of courses. Courses considered to have most likelihood of generating higher income for students in the future (e.g. Law and Medicine) are the most expensive and those least likely to generate higher income (e.g. Nursing and Arts) are the least expensive. Since 2007, HECS places are known as Commonwealth Supported Places (CSP). A student in a CSP is only entitled to study for a maximum of 7 years full-time (16 years part-time) at CSP rates. This is known as a Student Learning Entitlement (SLE). After that period the student has to take either a FEE-HELP loan (if available) or study at full-fee rates. If a student receives a HECS-HELP loan, the Commonwealth government pays the loan amount directly to the higher education provider on behalf of the student.

An alternative option is FEE-HELP which provides eligible fee-paying students with a loan to cover their tertiary education fees.

Commonwealth Supported Places (CSP) & Student Contribution (SC)
The Commonwealth government determines the number and allocation of undergraduate "Commonwealth Supported Places" (CSP) with each public higher education provider each year, through the Commonwealth Grant Scheme (CGS). A CSP is a higher education place for which the Commonwealth government makes a contribution to the higher education provider towards the cost of a student's education. The student makes a contribution towards the cost of education, known as the "Student Contribution" (SC). Commonwealth supported places are available to citizens of Australia and New Zealand and Australian permanent residents.

The majority of CSPs are managed through the Tertiary Admissions Centre (TACs) in each state or territory, although universities make the selections, deciding which students they will make offers to.

The allocation is usually based on secondary school results (through the ATAR scores), TAFE qualifications and previous university results.

The student contribution varies between courses, and is based on the expected earnings following a students' graduation, not the cost of providing the course. Higher education providers can set the student contribution level for each unit of study, up to a maximum level set by the government. It is said that, due to government underfunding of universities, universities almost always charge the highest level allowable.

EFTSL (Equivalent Full-time Student Load) is a measure of a full-time student’s annual study load in terms of minimum number of units (subjects) undertaken by a student in a particular semester, anything less is considered a part-time student.

Between 2012 and 2017, an eligible student who paid the entire or a part of the student contribution upfront received a 10% HECS discount on the amount paid (prior to 2012, the HECS discount was 20%). Only Australian citizens and permanent humanitarian visa holders were eligible for the up-front 10% HECS discount. The up-front discount was removed on 1 January 2017.

Total funding
The total funding available to institutions per equivalent full-time student is the combination of the student contribution (divided into 3 different amounts/bands) and the Commonwealth government contribution (divided into 8 different amounts/clusters). For 2017 these are:

Full fee-paying students
Full fee places for Australian undergraduate students were phased out in 2009 under reforms made by the Gillard government.

Other students may obtain a full fee place (FFP) if they do not receive a Commonwealth supported place, subject to meeting relevant qualifications. Most postgraduate courses do not have Commonwealth supported places available and therefore, all these students are full fee-paying. Fee-paying students are charged the full cost of their course, with no Commonwealth contribution.

Some fee-paying students can obtain loans under the Higher Education Loan Programme, called FEE-HELP loans, to cover all or part of their fees. This is available to Australian citizens, New Zealand citizens and permanent humanitarian visa holders. Undergraduate students who obtain these loans are charged a 20% loan fee on top of the amount borrowed. This does not apply to post graduate courses. Students are able to borrow a lifetime maximum FEE-HELP loan of $112,134 for medicine, dentistry and veterinary science programs and $89,706 for all other programs (adjusted for inflation). In 2005, FEE-HELP loans replaced the Open Learning Deferred Payment Scheme (OLDPS), the Postgraduate Education Loan Scheme (PELS) and the Bridging for Overseas-Trained Professionals Loan Scheme (BOTPLS).

OS-HELP
OS-HELP is a loan scheme to assist some undergraduate domestic students to undertake some, but not all, of their course of study overseas. Students are able to obtain a loan up to $6,470 (if the student will not be studying in Asia) or $7764 (if the student will be studying in Asia) for every six months, but can only receive a total of two loans throughout their lifetime. Unlike other loans in the HELP, the loan amount is paid directly to the student and the terms for the loans are set out by the tertiary providers.

As in the FEE-HELP loan scheme, a 20% fee applies on the amount borrowed. This 20% "administration fee" was removed for OS-HELP loans received after 1 January 2010.

SA-HELP
SA-HELP loan covers the SSAF (student services and amenities fee). The universities are not allowed to charge more than A$336 per year as SSAF.

HELP loan management
HELP debts do not attract interest (in the normal sense), but are instead indexed to the Consumer Price Index (CPI) on 1 June each year, based on the annual CPI to March of that year. The indexation rate applied on 1 June 2006 was 2.8% and 3.4% on 1 June 2007. Indexation applies to the part of the debt that has been unpaid for 11 months or more. Thus, indexation is calculated on the opening HELP debt balance on 1 July of the previous year plus any debt incurred in the first half of the current year (usually for first semester courses) less any compulsory and voluntary repayments, with bonus. Any HELP debt incurred on second semester courses (usually determined in June) will not be subject to indexation until the next year. After indexation, the new balance is rounded down to a whole dollar amount. Additionally, HELP debts are subject to a 25% fee which does not count towards a student’s HELP debt limit.

As of 1 January 2017 the Commonwealth Government removed the 5% voluntary repayment bonus on all HELP debt repayments.

If a person with an accumulated HELP debt dies, any compulsory repayment included on their income tax notice of assessment relating to the period prior to their death must be paid from their estate, but the remainder of their debt is cancelled.

Repayments
HELP debts are administered by the Australian Taxation Office and will be repaid compulsorily over time through the taxation system. If the HELP Repayment Income (HRI) of a person with a HELP debt exceeds a certain threshold, which for the 2014/15 financial year is $53,345, a compulsory payments will be deducted from the person's tax for the year. The HRI is the person's taxable income plus any net rental loss claimed against that taxable income and adding fringe benefits, reportable superannuation contributions and foreign income received, normally exempt from taxation.

Unlike marginal tax rates, the repayment rate applies on the full HRI, so that a person with a HRI below $45,881 in 2019/20 will not need to make a compulsory HELP repayment, but a person with a HRI of $80,000 would make a payment of $4,400. This is 5.5% of the HRI (not taxable income or the debt balance) of $80,000. The compulsory repayment amount cannot exceed the balance of the HELP debt.

The rates for compulsory repayment since 2006 have been:

It is also possible to make voluntary payments to further reduce the debt. Until 31 December 2004 voluntary payments over $500 earned a 15% bonus, from 1 January 2005 this was reduced to 10% and from 1 January 2012 this was reduced to 5%. From 1 January 2017 the Government removed the 5% repayment bonus.

title: Tertiary education fees in Australia


text:
Edward Canby

Edward Richard Sprigg Canby (November 9, 1817 – April 11, 1873) was a career United States Army officer and a Union general in the American Civil War. He served as a military governor after the war.

In 1861–1862, Canby commanded the Department of New Mexico, defeating the Confederate General Henry Hopkins Sibley at the Battle of Glorieta Pass, forcing him to retreat to Texas. At the war's end, he took the surrender of Generals Richard Taylor and Edmund Kirby Smith.

As commander of the Pacific Northwest in 1873, he was assassinated by Chief Kintpuash during peace talks with the Modoc, who were refusing to move from their California homelands.

Early life
Canby was born in Piatt's Landing, Kentucky, to Israel T. and Elizabeth (Piatt) Canby. He attended Wabash College, but transferred to the United States Military Academy, from which he graduated in 1839. He was commissioned a second lieutenant in the 2nd U.S. Infantry and served as the regimental adjutant.

Although he was often referred to as Edward Canby, a biographer has suggested that he was known as "Richard" during childhood and to some friends for most of his life. He was called "Sprigg" by fellow cadets at West Point, but during most of his career, he was generally referred to as E.R.S. Canby, sometimes signing his name "Ed. R.S. Canby."

Marriage and family
He married Louisa Hawkins Canby at Crawfordsville, Indiana, August 1, 1839. She came from a family of three sisters and a brother, with whom she remained close. The Canbys had one child, a daughter, who did not survive childhood.

Early military career
During his early career, Canby served in the Second Seminole War in Florida and saw combat during the Mexican–American War, where he received three brevet promotions, including to major for Contreras and Churubusco, and lieutenant colonel for Belén Gates. He also served at various posts, including Upstate New York and in the adjutant general's office in California from 1849 until 1851, covering the period of the territory's transition to statehood.

Against his wishes, he was assigned to what was supposed to be the civilian post of custodian of the California Archives from March 1850 until he left California in April 1851. The Archives included records of Spanish and Mexican governments in California, as well as Mission records and land titles. Evidently, Canby had some knowledge of the Spanish language, which was extremely useful as the government was trying to unravel land titles. (The Filson Historical Society in Louisville, Kentucky holds what appears to be a document written in Canby's hand in Spanish, in which he identifies himself as "Edwardo [sic] Ricardo S. Canby.")

Canby served in Wyoming and Utah (then both part of the Utah Territory) during the Utah War (1857–1858). During this period, he served on the panel of judges for the court martial of Captain Henry Hopkins Sibley. Sibley was acquitted. Subsequently, Canby wrote an endorsement for a teepee-like army tent which Sibley had adapted from the American Indian style.

Both officers were later assigned to New Mexico, where in 1860 Canby coordinated a campaign against the Navajo, commanding Sibley in a futile attempt to capture and punish Navajo for "depredations" against the livestock of settlers. The campaign ended in frustration, with Canby and Sibley rarely sighting Navajo raiders. Usually they saw the Navajo at a distance and never got close to them.

Civil War
At the start of the Civil War, Canby commanded Fort Defiance, New Mexico Territory. He was promoted to colonel of the 19th U.S. Infantry on May 14, 1861, and the following month commanded the Department of New Mexico. His former assistant Sibley resigned to join the Confederate Army, becoming a Brigadier General. Although Sibley's Army of New Mexico defeated Canby and his troops in February 1862 at the Battle of Valverde, Canby eventually forced the Confederates to retreat to Texas after the Union's strategic victory at the Battle of Glorieta Pass.

Immediately following this battle, Canby was promoted to brigadier general on March 31, 1862. Recombining the forces he had earlier divided, Canby set off in pursuit of the retreating Confederate forces, but he soon gave up the chase and allowed them to reach Texas. Shortly after the failure of the Confederate invasion of northern New Mexico, Canby was relieved of his command by Gen. James H. Carleton and reassigned to the east.

Canby's achievement in New Mexico had largely been in his planning an overall defensive strategy. He and his opponent, Sibley, both had limited resources. Though Canby was a little better supplied, he saw that defending the entire territory from every possible attack would stretch his forces too thinly. Realizing that Sibley had to attack along a river, especially since New Mexico was in the middle of a long drought, Canby made the best use of his forces by defending against only two possible scenarios: an attack along the Rio Grande and an attack by way of the Pecos and Canadian rivers. He could easily shift the latter defensive force to protect Fort Union if the enemy attacked by way of the Rio Grande, which they did. Canby persuaded the governors of both New Mexico and Colorado to raise volunteer units to supplement regular Federal troops; the Colorado troops proved helpful at both Valverde and Glorieta. In spite of occasional superior soldiering by Confederate troops and junior commanders, Sibley's sluggishness and vacillation in executing a plan with high risk led to an almost inevitable Confederate collapse.

After a period of clerical duty, Canby was assigned as "commanding general of the city and harbor of New York City" on July 17, 1863. This assignment followed the New York Draft Riots, which caused about 120 deaths and extensive property damage. He served until November 9, reviving the draft, and overseeing a prisoner of war camp in New York Harbor. He then went to work in the office of the Secretary of War, unofficially describing himself in correspondence as an "Assistant Adjutant General." Looking back on Canby's record, a 20th-century adjutant general, Edward F. Witsell, described Canby's position as "similar to that of an Assistant to the Secretary of the Army."

In May 1864, Canby was promoted to major general and relieved Nathaniel P. Banks of his command at Simmesport, Louisiana. He next was assigned to the Midwest, where he commanded the Military Division of Western Mississippi. He was wounded in the upper thigh by a guerrilla while aboard the gunboat USS Cricket on the White River in Arkansas near Little Island on November 6, 1864. Canby commanded the Union forces assigned to conduct the campaign against Mobile, Alabama in the spring of 1865. This culminated in the Battle of Fort Blakeley, which led to the fall of Mobile on April 12, 1865. Canby accepted the surrender of the Confederate forces under General Richard Taylor in Citronelle, on May 4, 1865, and those under General Edmund Kirby Smith west of the Mississippi River on May 26, 1865.

Canby was generally regarded as a great administrator, but he was criticized as a soldier. Ulysses S. Grant thought him not aggressive enough. At one time, Grant sent Canby an order to "destroy [the enemy's] railroads, machine-shops, &c." Ten days later, Grant reprimanded him for requesting men and materials to build railroads. "I wrote... urging you to... destroy railroads, machine-shops, &c., not to build them", Grant said. Canby could be a destroyer but appeared to prefer the role of builder. If someone had a question about army regulations or Constitutional law affecting the military, Canby was the man to see. Grant came to appreciate this in peace time, once complaining vigorously when President Andrew Johnson proposed to assign Canby away from the capital where Grant considered him irreplaceable.

Central Lunatic Asylum for Colored Insane
In April 1869, Secretary of War Edwin Stanton appointed General Canby as military governor of Virginia. Soon after Canby arrived in Richmond, he confiscated each of the medical facilities in the city and converted them for use by the Union Army. In the next several months, Canby was made aware of the critical medical and economic plight of thousands of formerly enslaved blacks in the state uprooted by the Civil War. Canby had to decide how to provide blacks access to health and mental health services without violating the racial pecking order that existed in the South. One area in dispute was whether blacks would be allowed admission to the state's existing mental asylums at Williamsburg and Staunton. Racial integration of these two asylums had been debated in the legislature and the psychiatric community for over a decade. Dr. John Galt, superintendent of Eastern Lunatic Asylum at Williamsburg believed that free blacks and whites could be treated medically in the same facility as he had demonstrated. However, Dr. Francis Stribling, superintendent of Western Lunatic Asylum at Staunton refused to admit either free or enslaved blacks to his institution. Following the death of Galt, Stribling became chair of an asylum planning committee that advised Canby and the Freedman's Bureau on a permanent admission policy for the black population. Stribling proposed that Virginia should construct a separate asylum for the admission and treatment of blacks with lunacy. Canby accepted his recommendation and included it as the basis of his military order number 136, published in December 1869. Canby's order required continued utilization of a rented annex at Howard's Grove Hospital as the temporary psychiatric hospital for blacks until the state of Virginia could decide whether to maintain and expand it or construct a new facility. In June 1870 the Virginia legislature accepted ownership of the Central Lunatic Asylum for Colored Insane, the first standalone facility in the United States. It remained located at the Howard's Grove site until 1885, when a new facility was constructed in Dinwiddie County some 40 miles south of Richmond and renamed Central State Hospital. Canby should be credited with creating the first racially segregated mental hospital in the US for African Americans. The hospital remained segregated by race until passage of the Civil Rights Act of 1964.

Post-war assignments
After the war, Canby served as commander of various military departments during Reconstruction, as the government tried to manage dramatic social changes while securing peace. He commanded Louisiana from 1864 to May 1866. He was next assigned as commander of the Department of Washington, which consisted of Delaware, Maryland, the District of Columbia, and Alexandria and Fairfax counties in Virginia, from June 1866 until August 1867. He was assigned to the command of the Second Military District, comprising North and South Carolina. In August 1868, he briefly resumed command in Washington.

He was assigned to the Fifth Military District in November, where he focused primarily on the reconstruction of Texas. He left Texas for Virginia, the First Military District, in April 1869, serving there until July 1870. Each of these postings occurred during Reconstruction and put Canby at the center of conflicts between Republicans and Democrats, whites and blacks, state and federal governments. New state legislatures were writing constitutions, and the social climate was highly volatile, with insurgent attacks against freedmen and Republicans on the rise in numerous areas. Many of his districts had Ku Klux Klan chapters, which the US government was not able to suppress until the early 1870s. Canby sometimes alienated one side or the other and often both. Charles W. Ramsdell called Canby "vigorous and firm, but just." Even political opponents, such as Jonathan Worth, governor of North Carolina, admitted that Canby was sincere and honest.

Final assignment and death
In August 1872, Canby was posted to command the Pacific Northwest. He soon faced problems with the Modoc tribe, who had historically lived in Northern California. Forced to remove to a reservation in Oregon which they had to share with their historical enemies, the Klamath tribe, they had pled with the US government to return to California. When the US refused, the Modoc left the reservation and returned home. In 1872, the US Army went to fight against them to force them back and the Modoc War broke out. The Modoc, entrenched in Captain Jack's Stronghold south of Tule Lake, resisted army attacks and fought US forces to a stalemate. General Canby had received conflicting orders from Washington as to whether to make peace or war on the Modoc. As war was not working, the US government authorized a peace commission and assigned Canby a key position on it. There were many lines of communication between the Modoc and whites. At one point, someone told the Modoc leader Captain Jack that the governor of Oregon intended to hang nine Modoc, apparently without trial, as soon as they surrendered. The Modoc broke off scheduled talks; Canby was angered by the rumors and their action, as he believed that his federal authority trumped the governor's and made the threat irrelevant.

On April 11, 1873, after months of false starts and aborted meetings, Canby went to another parley, unarmed and with some hope of final resolution. Judge Elijah Steele of Yreka, California wrote later that when he warned Canby that the Modoc were volatile and he was at risk, Canby replied, "I believe you are right, Mr. Steele, and I shall regard your advice, but it would not be very well for the general in command to be afraid to go where the peace commissioners would venture." The peace talks were held midway between the army encampment and Captain Jack's stronghold near Tule Lake. It was discovered later that two members of Canby's party brought concealed weapons and Modoc warriors were also armed. According to Jeff C. Riddle, the Modoc son of the US interpreter and the author of Indian History of the Modoc War (1914), the Modoc had plotted before the meeting to kill Canby and the other commissioners, as they believed peace was not possible. They were determined to "fight until we die." (He was the son of Winema and Frank Riddle.) Captain Jack had been reluctant to agree to the killings, believing it "coward's work", but was pressured by other warriors to agree. He insisted on being given another chance to ask Canby to "give us a home in our country." When Canby said he did not have the authority to make such a promise, Captain Jack attacked the general. With Ellen's Man, one of his lieutenants, he shot Canby twice in the head and cut his throat. The Modoc also killed Reverend Eleazar Thomas, a peace commissioner, and wounded others in the party.

Aftermath
Following Canby's death, national outrage was expressed against the Modoc. Eastern newspapers called for blood vengeance, except for one in Georgia, which headlined the story: "Captain Jack and Warriors Revenge the South By Murdering General Canby, One of Her Greatest Oppressors." E.C. Thomas, son of the murdered peace commissioner, recognized the inevitability of reprisals for the killings, but said: "To be sure, peace will come through war, but not by extermination." Eventually, Captain Jack (Kintpuash), Boston Charley, Schonchin John, Black Jim, Barncho, and Sloluck were tried for "murder in violation of the laws of war", convicted, and sentenced to death. President Grant commuted the sentences of Barncho and Sloluck to life imprisonment. The other condemned men were all executed on October 3, 1873. The surviving Modoc were sent to reservations. Barncho died in prison in 1875, and Sloluck was released from prison in 1878.

The killing of Canby, and the Great Sioux War, undermined public confidence in President Grant's peace policy, according to the historian Robert Utley. There was growing public sentiment for full defeat of the American Indians.

After memorial services were performed on the West Coast, Canby's body was returned to Indiana and buried in Crown Hill Cemetery, Indianapolis, Indiana on May 23, 1873. At least four Union generals attended his funeral there: William Tecumseh Sherman, Philip Sheridan, Lew Wallace, and Irvin McDowell, and the latter two served among the pall bearers. A reporter noted that, although the funeral procession was generally reserved, "more than once, expressions of hatred toward the Modoc" marred the silence.

Legacy and honors

 * 1871, he was awarded an honorary degree by Wesleyan University of Middletown, Connecticut.
 * 1875, Fort Canby, a coast defense installation guarding the entrance to the Columbia River, is named in his honor.
 * 1880s, Canby's Cross was erected in his honor near the site of the peace talks, in the area later designated as the Lava Beds National Monument.
 * The towns of Canby in Clackamas County, Oregon, Canby in Yellow Medicine County, Minnesota, and Canby in Modoc County, California, are named for him.
 * Canby, Oregon's annual July 4th celebration used to be called General Canby Days, with activities including a pancake breakfast, car show, parade and music.

Cultural depictions
The 1954 Western Drum Beat depicted the 1873 Modoc War and Canby's death at the hands of Captain Jack at the peace conference. Canby was played by Warner Anderson and Captain Jack by Charles Bronson.

Col. Canby is referred to in 1966 movie "The Good, the Bad and the Ugly" as a Confederate column is retreating through town.

title: Edward Canby


text:
Talk:Collective trauma

Wiki Education Foundation-supported course assignment
This article was the subject of a Wiki Education Foundation-supported course assignment, between 9 September 2021 and 18 November 2021. Further details are available on the course page. Student editor(s): Mistry Shivani.

Above undated message substituted from Template:Dashboard.wikiedu.org assignment by PrimeBOT (talk) 18:01, 16 January 2022 (UTC)

July 2003
The last paragraph of this article, which was written 10 days after 9/11/2001, could do with updating. Does anyone feel in the position to write about collective trauma due to 9/11 two years down the line? Pete 10:48 9 Jul 2003 (UTC)

Psychological additions to this page
Hello,

As part of a university project, I have picked the collective trauma article to add on too. I plan to add onto the global impacts but with a more psychological and neurological focus of specific events, but this could be its own subheading about neurological impacts. I was also planning to add on a part about epigenetics and the biological effects of collective trauma. Another focus could be the PTSD aftermath and mental health links.

Thank you!

MU0212 (talk) 14:00, 24 January 2022 (UTC)

Lead too long?
I am just wondering why it is being flagged that the lead is too long, when it is only two sentences. Is there a way to remove that as one of the issues in this article or is the problem something else? MU0212 (talk) 13:47, 15 February 2022 (UTC)

Thomas Hübl is a problematic source
The last part of this article builds heavily on Thomas Hübl. This is problematic because:

 * The quotes are not academic sources, not peer reviewed.
 * The sources are his book, a blog post, and a TED talk.
 * Thomas Hübl is not an academic.
 * He is a "spiritual" or religous leader with a huge following.
 * He is also not a non-profit leader, while he certainly started a non-profit.
 * Thomas Hübl ownes a business that makes a lot of money with his teaching. Most of the mention here needs to be considered free marketing.
 * Without being able to cite any academic criticism of his work, I know from psychologist friends, that his work is diametric against some academic research about trauma.

I would like to see this topic discussed by psychologist or other academics in a broader way. How can this discussion be initiated on Wikipedia? 62.96.232.178 (talk) 11:30, 17 November 2023 (UTC)

Merge with historical trauma
There seems to be significant overlap between these two articles, although I'm not an expert on this at all. Alexanderkowal (talk) 09:37, 20 May 2024 (UTC)

title: Talk:Collective trauma


text:
Ruddy duck

The ruddy duck (Oxyura jamaicensis) is a duck from North America and one of the stiff-tailed ducks. The genus name is derived from Ancient Greek oxus, "sharp", and oura, "tail", and jamaicensis is "from Jamaica".

Taxonomy
The ruddy duck was formally described in 1789 by the German naturalist Johann Friedrich Gmelin in his revised and expanded edition of Carl Linnaeus's Systema Naturae. He placed it with the other ducks, geese and swans in the genus Anas and coined the binomial name Anas jamaicensis. Gmelin based his description on the "Jamaica shoveler" that had been described in 1785 by the English ornithologist John Latham from a specimen that he had received from Jamaica. The ruddy duck is now placed with five other species in the genus Oxyura that was introduced in 1828 by the French naturalist Charles Lucien Bonaparte. The genus name is derived from Ancient Greek oxus, meaning "sharp", and oura meaning "tail". The specific epithet jamaicensis means "from Jamaica". The Andean duck was formerly considered to be conspecific with the ruddy duck but with the two species split, the ruddy duck is monotypic: no subspecies are recognised.

Description
These are small, compact ducks with stout, scoop-shaped bills, and long, stiff tails they often hold cocked upward. They have slightly peaked heads and fairly short, thick necks. Male ruddy ducks have blackish caps that contrast with bright white cheeks. In summer, they have rich chestnut bodies with bright blue bills. In winter, they are dull gray-brown above and paler below with dull gray bills. Females and first-year males are brownish, somewhat like winter males but with a blurry stripe across the pale cheek patch. In flight, ruddy ducks show solidly dark tops of the wings.

Breeding and habits
Their breeding habitat is marshy lakes and ponds. They nest in dense marsh vegetation near water. The female builds the nest out of grass, locating it in tall vegetation to hide it from predators. A typical brood contains 5 to 15 ducklings. Pairs form each year.

They are migratory and winter in coastal bays and unfrozen lakes and ponds.

These birds dive and swim underwater. They mainly eat seeds and roots of aquatic plants, aquatic insects and crustaceans.

Invasive species/Cull
As a result of escapes from wildfowl collections in the late 1950s, they became established in Great Britain, from where they spread into Europe. This duck's aggressive courting behavior and willingness to interbreed with the endangered native white-headed duck (Oxyura leucocephala), of southern Europe, caused concern amongst Spanish conservationists. Due to this, a controversial scheme to extirpate the ruddy duck as a British breeding species started; there have also been culling attempts in other European countries.

By March 2012 a culling program in the UK, supported by the RSPB, had killed 6,500, at a cost of £5m (£769 per bird). In 2003 the BBC had reported the cost of killing each bird at £915. In 2012 Lee Evans, founder of the British Birding Association, claimed "The cull cannot succeed now. There are hundreds of ruddy ducks on the continent which will not be killed so the birds will continue to breed. There's never been any proof, anyway, that the British population has ever interbred with the Spanish ducks". By early 2014, the cull had reduced the British population to about 20–100, down from a peak of about 5,500 in 2000. According to Animal Aid, in the UK the cost of hunting down the last few ruddy ducks was £3,000 per bird. They advised "If you see one, don’t tell anyone. Even bird groups will tell the authorities and those birds may be killed".

In Europe, the Ruddy duck is included since 2016 in the list of Invasive Alien Species of Union concern (the Union list). This implies that this species cannot be imported, bred, transported, commercialized, or intentionally released into the environment in the whole of the European Union.

title: Ruddy duck


text:
Canby's Cross

Canby's Cross is located in Lava Beds National Monument, about 3 miles south of Tule Lake, and 5 miles south-southwest of the town of Tulelake, California. It was erected to commemorate General Canby's death at a peace gathering. General Canby was shot in the face by Captain Jack of the Modoc tribe, who was later hanged for the killing. The cross is registered as a California Historical Landmark.

The cross reads: Gen Canby USA was murdered here by the Modocs April 11, 1873.

The elevation of Canby's Cross is 4058 ft (1237 m).

title: Canby's Cross


text:
Evanescent field

In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillating charges and currents). Even when there is a propagating electromagnetic wave produced (e.g., by a transmitting antenna), one can still identify as an evanescent field the component of the electric or magnetic field that cannot be attributed to the propagating wave observed at a distance of many wavelengths (such as the far field of a transmitting antenna).

A hallmark of an evanescent field is that there is no net energy flow in that region. Since the net flow of electromagnetic energy is given by the average Poynting vector, this means that the Poynting vector in these regions, as averaged over a complete oscillation cycle, is zero.

Use of the term
In many cases one cannot simply say that a field is or is not "evanescent" – having the Poynting vector average to zero in some direction (or all directions). In most cases where they exist, evanescent fields are simply thought of and referred to the same as all other electric or magnetic fields involved, without any special recognition of those fields' evanescence. The term's use is mostly limited to distinguishing a part of a field or solution in those cases where one might only expect the fields of a propagating wave.

For instance, in the illustration at the top of the article, energy is indeed carried in the horizontal direction. However, in the vertical direction, the field strength drops off exponentially with increasing distance above the surface. This leaves most of the field concentrated in a thin boundary layer very close to the interface; for that reason, it is referred to as a surface wave. However, despite energy flowing horizontally, along the vertical there is no net propagation of energy away from (or toward) the surface, so that one could properly describe the field as being "evanescent in the vertical direction". This is one example of the context dependence of the term.

Everyday electronic devices and electrical appliances are surrounded by large fields which are evanescent; their operation involves alternating voltages (producing an electric field between them) and alternating currents (producing a magnetic field around them) which are expected to only carry power along internal wires, but not to the outsides of the devices. Even though the term "evanescent" is not mentioned in this ordinary context, the appliances' designers still may be concerned with maintaining evanescence, in order to prevent or limit production of a propagating electromagnetic wave, which would lead to radiation loss, since a propagating wave "steals" its power from the circuitry or donates unwanted interference.

The term "evanescent field" does arise in various contexts where a propagating electromagnetic wave is involved (even if confined). The term then differentiates electromagnetic field components that accompany the propagating wave, but which do not themselves propagate. In other, similar cases, where a propagating electromagnetic wave would normally be expected (such as light refracted at the interface between glass and air), the term is invoked to describe that part of the field where the wave is suppressed (such as light traveling through glass, impinging on a glass-to-air interface but beyond the critical angle).

Although all electromagnetic fields are classically governed according to Maxwell's equations, different technologies or problems have certain types of expected solutions, and when the primary solutions involve wave propagation the term evanescent is frequently applied to field components or solutions which do not share that property.

For instance, the propagation constant of a hollow metal waveguide is a strong function of frequency (a dispersion relation). Below a certain frequency (the cut-off frequency) the propagation constant becomes an imaginary number. A solution to the wave equation having an imaginary wavenumber does not propagate as a wave but falls off exponentially, so the field excited at that lower frequency is considered evanescent. It can also be simply said that propagation is "disallowed" for that frequency.

The formal solution to the wave equation can describe modes having an identical form, but the change of the propagation constant from real to imaginary as the frequency drops below the cut-off frequency totally changes the physical nature of the result. The solution may be described as a "cut-off mode" or an "evanescent mode";  while a different author will just state that no such mode exists. Since the evanescent field corresponding to the mode was computed as a solution to the wave equation, it is often discussed as being an "evanescent wave" even though its properties (such as carrying no energy) are inconsistent with the definition of wave.

Although this article concentrates on electromagnetics, the term evanescent is used similarly in fields such as acoustics and quantum mechanics, where the wave equation arises from the physics involved. In these cases, solutions to the wave equation resulting in imaginary propagation constants are likewise called "evanescent", and have the essential property that no net energy is transferred, even though there is a non-zero field.

Evanescent wave applications
In optics and acoustics, evanescent waves are formed when waves traveling in a medium undergo total internal reflection at its boundary because they strike it at an angle greater than the critical angle. The physical explanation for the existence of the evanescent wave is that the electric and magnetic fields (or pressure gradients, in the case of acoustical waves) cannot be discontinuous at a boundary, as would be the case if there was no evanescent wave field. In quantum mechanics, the physical explanation is exactly analogous—the Schrödinger wave-function representing particle motion normal to the boundary cannot be discontinuous at the boundary.

Electromagnetic evanescent waves have been used to exert optical radiation pressure on small particles to trap them for experimentation, or to cool them to very low temperatures, and to illuminate very small objects such as biological cells or single protein and DNA molecules for microscopy (as in the total internal reflection fluorescence microscope). The evanescent wave from an optical fiber can be used in a gas sensor, and evanescent waves figure in the infrared spectroscopy technique known as attenuated total reflectance.

In electrical engineering, evanescent waves are found in the near-field region within one third of a wavelength of any radio antenna. During normal operation, an antenna emits electromagnetic fields into the surrounding nearfield region, and a portion of the field energy is reabsorbed, while the remainder is radiated as EM waves.

Recently, a graphene-based Bragg grating (one-dimensional photonic crystal) has been fabricated and demonstrated its competence for excitation of surface electromagnetic waves in the periodic structure using a prism coupling technique.

In quantum mechanics, the evanescent-wave solutions of the Schrödinger equation give rise to the phenomenon of wave-mechanical tunneling.

In microscopy, systems that capture the information contained in evanescent waves can be used to create super-resolution images. Matter radiates both propagating and evanescent electromagnetic waves. Conventional optical systems capture only the information in the propagating waves and hence are subject to the diffraction limit. Systems that capture the information contained in evanescent waves, such as the superlens and near field scanning optical microscopy, can overcome the diffraction limit; however these systems are then limited by the system's ability to accurately capture the evanescent waves. The limitation on their resolution is given by
 * $$ k \propto \frac{1}{d} \ln \frac{1}{\delta},$$

where $$k$$ is the maximal wave vector that can be resolved, $$d$$ is the distance between the object and the sensor, and $$\delta$$ is a measure of the quality of the sensor.

More generally, practical applications of evanescent waves can be classified as (1) those in which the energy associated with the wave is used to excite some other phenomenon within the region of space where the original traveling wave becomes evanescent (for example, as in the total internal reflection fluorescence microscope) or (2) those in which the evanescent wave couples two media in which traveling waves are allowed, and hence permits the transfer of energy or a particle between the media (depending on the wave equation in use), even though no traveling-wave solutions are allowed in the region of space between the two media. An example of this is wave-mechanical tunnelling, and is known generally as evanescent wave coupling.

Total internal reflection of light

For example, consider total internal reflection in two dimensions, with the interface between the media lying on the x axis, the normal along y, and the polarization along z. One might expect that for angles leading to total internal reflection, the solution would consist of an incident wave and a reflected wave, with no transmitted wave at all, but there is no such solution that obeys Maxwell's equations. Maxwell's equations in a dielectric medium impose a boundary condition of continuity for the components of the fields Eundefined, Hundefined, Dy, and By. For the polarization considered in this example, the conditions on Eundefined and By are satisfied if the reflected wave has the same amplitude as the incident one, because these components of the incident and reflected waves superimpose destructively. Their Hx components, however, superimpose constructively, so there can be no solution without a non-vanishing transmitted wave. The transmitted wave cannot, however, be a sinusoidal wave, since it would then transport energy away from the boundary, but since the incident and reflected waves have equal energy, this would violate conservation of energy. We therefore conclude that the transmitted wave must be a non-vanishing solution to Maxwell's equations that is not a traveling wave, and the only such solutions in a dielectric are those that decay exponentially: evanescent waves.

Mathematically, evanescent waves can be characterized by a wave vector where one or more of the vector's components has an imaginary value. Because the vector has imaginary components, it may have a magnitude that is less than its real components.

For the plane of incidence as the $$ xy $$ plane at $$ z = 0 $$ and the interface of the two mediums as the $$ xz $$ plane at $$ y=0 $$, the wave vector of the transmitted wave has the form
 * $$ \mathbf{k_t} \ = \ k_y \hat{\mathbf{y}} + k_x \hat{\mathbf{x}} $$

with $$ k_x = k_t \sin \theta_t $$ and $$ k_y = k_t \cos \theta_t $$, where $$ k_t $$ is the magnitude of the wave vector of the transmitted wave (so the wavenumber), $$ \theta_t $$ is the angle of refraction, and $$ \hat{\mathbf{x}} $$ and $$ \hat{\mathbf{y}} $$ are the unit vectors along the $$ x $$ axis direction and the $$ y $$ axis direction respectively.

By using the Snell's law $$ n_i \sin \theta_i = n_t \sin \theta_t $$ where $$ n_i $$, $$ n_t $$, and $$ \theta_i $$ are the refractive index of the medium where the incident wave and the reflected wave exist, the refractive index of the medium where the transmitted wave exists, and the angle of incidence respectively,
 * $$ k_y =k_t \cos \theta_t = \pm k_t \left (1 - \frac{\sin^2 \theta_i}{n_{ti}^2}\right)^{1/2} $$.

with $ n_{ti} = \frac{n_t}{n_i} $.

If a part of the condition of the total internal reflection as $$ \sin \theta_i > \sin \theta_c = n_{ti} $$, is satisfied, then
 * $$ k_y = \pm i k_t \left( \frac{\sin^2 \theta_i}{n_{ti}^2} - 1\right)^{1/2} = \pm i \alpha $$.

If the polarization is perpendicular to the plane of incidence (along the $$ z $$ direction), then the electric field of any of the waves (incident, reflected, or transmitted) can be expressed as

 * $$ \mathbf{E}(\mathbf{r},t) = \operatorname{Re} \left \{ E(\mathbf{r}) e^{ i \omega t } \right \} \mathbf{\hat{z}} $$

where $$\mathbf{\hat{z}}$$ is the unit vector in the $$ z $$ axis direction.

By assuming plane waves as $$ E(\mathbf{r}) = E_0 e^{-i\mathbf{k}\cdot \mathbf{r}} $$, and substituting the transmitted wave vector $$ \mathbf{k_t} $$ into $$ \mathbf{k} $$, we find for the transmitted wave:

 * $$ E(\mathbf{r}) =  E_o e^{-i ( -i \alpha y + \beta x ) }  =  E_o e^{-\alpha y - i \beta x } $$

where $ \alpha = k_t \left( \frac{\sin^2 \theta_i}{n_{ti}^2} - 1\right)^{1/2} $ is the attenuation constant, and $$ \beta = k_x $$ is the phase constant. $$ +i \alpha $$ is ignored since it does not physically make sense (the wave amplification along y the direction in this case).

Evanescent-wave coupling
Especially in optics, evanescent-wave coupling refers to the coupling between two waves due to physical overlap of what would otherwise be described as the evanescent fields corresponding to the propagating waves.

One classical example is frustrated total internal reflection (FTIR) in which the evanescent field very close (see graph) to the surface of a dense medium at which a wave normally undergoes total internal reflection overlaps another dense medium in the vicinity. This disrupts the totality of the reflection, diverting some power into the second medium.

Coupling between two optical waveguides may be effected by placing the fiber cores close together so that the evanescent field generated by one element excites a wave in the other fiber. This is used to produce fiber-optic splitters and in fiber tapping. At radio (and even optical) frequencies, such a device is called a directional coupler. The device is usually called a power divider in the case of microwave transmission and modulation.

Evanescent-wave coupling is synonymous with near field interaction in electromagnetic field theory. Depending on the nature of the source element, the evanescent field involved is either predominantly electric (capacitive) or magnetic (inductive), unlike (propagating) waves in the far field where these components are connected (identical phase, in the ratio of the impedance of free space). The evanescent wave coupling takes place in the non-radiative field near each medium and as such is always associated with matter; i.e., with the induced currents and charges within a partially reflecting surface. In quantum mechanics the wave function interaction may be discussed in terms of particles and described as quantum tunneling.

Applications
Evanescent wave coupling is commonly used in photonic and nanophotonic devices as waveguide sensors or couplers (see e.g., prism coupler).

Evanescent wave coupling is used to excite, for example, dielectric microsphere resonators.

Evanescent coupling, as near field interaction, is one of the concerns in electromagnetic compatibility.

Coupling of optical fibers without loss for fiber tapping.

Evanescent wave coupling plays a major role in the theoretical explanation of extraordinary optical transmission.

Evanescent wave coupling is used in powering devices wirelessly.

A total internal reflection fluorescence microscope uses the evanescent wave produced by total internal reflection to excite fluorophores close to a surface. This is useful when surface properties of biological samples need to be studied.

title: Evanescent field