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Dianic Wicca

Dianic Wicca, also known as Dianic Witchcraft, is a modern pagan goddess tradition focused on female experience and empowerment. Leadership is by women, who may be ordained as priestesses, or in less formal groups that function as collectives. While some adherents identify as Wiccan, it differs from most traditions of Wicca in that only goddesses are honored (whereas most Wiccan traditions honor both female and male deities).

While there is more than one tradition known as Dianic, the most widely known is the female-only variety, with the most prominent tradition thereof founded by Zsuzsanna Budapest in the United States in the 1970s. It is notable for its worship of a single, monotheistic Great Goddess (with all other goddesses - of all cultures worldwide - seen as "aspects" of this goddess) and a focus on egalitarian matriarchy. While the tradition is named after the Roman goddess Diana, Dianics worship goddesses from many cultures, within the Dianic Wiccan ritual framework. Diana, (considered correlate to the Greek Artemis) "is seen as representing a central mythic theme of woman-identified cosmology. She is the protector of women and of the wild, untamed spirit of nature."

The Dianic Wiccan belief and ritual structure is an eclectic combination of elements from British Traditional Wicca, Italian folk-magic as recorded by Charles Leland in Aradia, New Age beliefs, and folk magic and healing practices from a variety of different cultures.

Beliefs and practices
Dianic Wiccans of the Budapest lineage worship the Goddess, who they see as containing all goddesses, from all cultures; she is seen as the source of all living things and containing all that is within her. "While Diana does have a triple aspect, it is in Her aspect as Virgin Huntress that She guides Her daughters to wholeness. She is “virgin” in the ancient sense of “She Who Is Whole Unto Herself.” The ancient meaning of “virgin” described a woman who was unmarried, autonomous, belonging solely to herself. The original meaning of this word was not attached to a sexual act with a man. Diana/Artemis did not associate herself or consort with men, which is why these Goddesses are often understood to be lesbian."

Dianic covens practice magic in the form of meditation and visualization in addition to spell work. They focus especially on healing themselves from the wounds of the patriarchy while affirming their own womanhood.

Rituals can include reenacting religious and spiritual lore from a female-centered standpoint, celebrating the female body, and mourning society's abuses of women. The practice of magic is rooted in the belief that energy or 'life force' can be directed to enact change. However it is important to note that rituals are often improvised to suit individual or group needs and vary from coven to coven. Some Dianic Wiccans eschew manipulative spellwork and hexing because it goes against the Wiccan Rede. However, many other Dianic witches (notably Budapest) do not consider hexing or binding of those who attack women to be wrong, and actively encourage the binding of rapists.

Differences from mainstream Wicca
Like other Wiccans, Dianics may form covens, attend festivals, celebrate the eight major Wiccan holidays, and gather on Esbats. They use many of the same altar tools, rituals, and vocabulary as other Wiccans. Dianics may also gather in less formal Circles. The most noticeable difference between the two are that Dianic covens of Budapest lineage are composed entirely of women. Central to feminist Dianic focus and practice are embodied Women's Mysteries - the celebrations and honoring of the female life cycle and its correspondences to the Earth's seasonal cycle, healing from internalized oppression, female sovereignty and agency. Another marked difference in cosmology from other Wiccan traditions is rejecting the concept of duality based in gender stereotypes.

When asked why "men and gods" are excluded from her rituals, Budapest stated:

"It's the natural law, as women fare so fares the world, their children, and that's everybody. If you lift up the women you have lifted up humanity. Men have to learn to develop their own mysteries. Where is the order of Attis? Pan? Zagreus? Not only research it, but then popularize it as well as I have done. Where are the Dionysian rites? I think men are lazy in this aspect by not working this up for themselves. It's their own task, not ours."

- during a 2007 interview

Sociological studies have shown that there is therapeutic value inherent in Dianic ritual. Healing rituals to overcome personal trauma and raise awareness about violence against women have earned comparisons to the female-centered consciousness-raising groups of the 1960s and 70s. Some Dianic groups develop rituals specifically to confront gendered personal trauma, such as battery, rape, incest, and partner abuse. In one ethnographic study of such a ritual, women shifted their understanding of power from the hands of their abusers to themselves. It was found that this ritual had improved self-perception in participants in the short-term, and that the results could be sustained with ongoing practice. Dianic Wicca developed from the Women's Liberation Movement and some covens traditionally compare themselves with radical feminism. Dianics pride themselves on the inclusion of lesbian and bisexual members in their groups and leadership. It is a goal within many covens to explore female sexuality and sensuality outside of male control, and many rituals function to affirm lesbian sexuality, making it a popular tradition for lesbians and bisexuals. Some covens exclusively consist of same-sex oriented women and advocate lesbian separatism. Ruth Barrett writes,

For other lesbian Dianics, as well as heterosexual and bisexual Dianics, excluding males from participation in ritual is not born from a rejection of males but rather an embracing of women’s unique biological rites of passage and how living in female body in a patriarchal world informs and effects our lives.

Many women choose Dianic separatist ritual simply because of the joy, fun, pleasure, feeling of safety, and value which they derive from being in a exclusively female space with other like-minded women.

History
Aradia, or the Gospel of the Witches claims that ancient Diana, Aphrodite, Aradia, and Herodias cults linked to the Sacred Mysteries are the origin of the all-female coven and many witch myths as we know them.

Z Budapest's branch of Dianic Wicca began on the Winter Solstice of 1971, when Budapest led a ceremony in Hollywood, California. Self-identifying as a "hereditary witch," and claiming to have learned folk magic from her mother,  Budapest is frequently considered the mother of modern Dianic Wiccan tradition. Dianic Wicca itself is named after the Roman goddess of the same name. Ruth Rhiannon Barrett was ordained by Z Budapest in 1980 and inherited Budapest's Los Angeles ministry. This community continues through Circle of Aradia, a grove of Temple of Diana, Inc.

Denominations and related traditions

 * Traditions derived from Zsuzsanna Budapest - Female-only covens run by priestesses trained and initiated by Budapest.
 * Independent Dianic witches - who may have been inspired by Budapest, her published work (such as The Holy Book of Women's Mysteries) or other woman's spirituality movements, and who emphasize independent study and self-initiation.

McFarland Dianic
McFarland Dianic is a Neopagan tradition of goddess worship founded by Morgan McFarland and Mark Roberts which, despite the shared name, has a different theology and structure than the women-only groups. In most cases, the McFarland Dianics accept male participants. McFarland largely bases their tradition on the work of Robert Graves and his book The White Goddess. While some McFarland covens will initiate men, the leadership is limited to female priestesses. Like the women-only Dianic traditions, "McFarland Dianic covens espouse feminism as an all-important concept." They consider the decision whether to include or exclude males as "solely the choice of [a member coven's] individual High Priestess."

Criticism for transphobia
Dianic Wicca has been criticised by elements in the Neopagan community for being transphobic. In February 2011, Zsuzsanna Budapest conducted a ritual with the Circle of Cerridwen at PantheaCon for "genetic women only" from which she barred trans women as well as men. This caused a backlash that led many to criticize Dianic Wicca as an inherently transphobic lesbian-separatist movement. The Los Angeles Times wrote that:

"Talia Bettcher, a professor of philosophy at Cal State L.A., said this trans exclusionary stance is common — though not universal — among women like Budapest who were grounded in lesbian separatism, a political vision that originated in the ’70s. [...] The same defiance that made Budapest a feminist hero in the ’70s has made her a divisive figure for many modern witches today."

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

title: Dianic Wicca


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Talk:Diffraction

Diffractive Optics
"Diffractive Optics" redirects to this article. Diffractive Optics is a technique for making smaller and lighter lenses. It has nothing whatsover to do with the content of this article. 86.147.237.233 (talk) 12:40, 17 March 2012 (UTC)
 * Actually, these lighter lenses you refer to work by the principles of diffraction described in this article. But if you'd like to start an article on diffractive optics, that would probably be OK, too.  Dicklyon (talk) 18:46, 17 March 2012 (UTC)

 * Are you sure about that? I was under the impression that "Diffractive Optics" was just a Canon marketing term for Fresnel lenses which don't actually work by diffraction.  The DO lenses contain Fresnel lenses.  Even so, the article doesn't mention the lens construction so my point stands. 86.147.237.233 (talk) 14:32, 22 March 2012 (UTC)

 * I have seen reference to diffractive optics being a Canon marketing term in several reviews. Diffraction in a camera lens is a bad thing to be avoided where possible.  Every camera lens suffers from the degrading effects of diffraction to some extent.  The light travelling through a lens always diffracts around the aperture blades softening the image to some extent.  As the aperture gets smaller, the diffracted light becomes a larger proportion of the total light throughput detracting from the sharpness of the image.  In general, lenses usually give their sharpest image somewhere around f/8.  When lenses are constructed out of Fresnel lenses, the steps between the lens segments introduces several further edges around which light diffracts, thus diffractive optic lenses have a lower optical performance. 109.145.22.224 (talk) 12:31, 5 April 2012 (UTC)

Propagation of a laser beam
In this section, it is mentioned "Hence, the smaller the output beam, the quicker it diverges". Here, does "smaller" mean a smaller size or something else? Hopefully someone can help me with this question, thanks! AlexHe34 (talk) 15:14, 30 March 2012 (UTC)

 * When light is incident on an aperture, the angle into which it is diffracted is inversely proportional to the size of the aperture. One can view the output beam of a laser as being defined by the output aperture of the system.  This is much smaller in a semiconductor laer than in, say a He-Ne laser, hence the semiconductor beam spreads out much more than a He-Ne one.  It can, of course be collimated (converted into a  beam which spreads only very slowly), as in a laser pointer, by using a lens.Epzcaw (talk) 20:50, 22 July 2012 (UTC)

I have fixed few other things here. Laser mirror is a resonator, not an aperture. And the laser beam is not always a fundamental mode (in fact, almost never). — Preceding unsigned comment added by Khrapkorr (talk • contribs) 20:30, 10 February 2015 (UTC)

Double-slit diffraction
You write: "Thomas Young performed a celebrated experiment in 1803 demonstrating interference from two closely spaced slits.[10]"

If you actually read the Bakerian lecture you reference [10] you will see that he does not discuss the celebrated double-slit experiment, nor is the diagram you print in that lecture. the only place Young mentions the experiments is in his popular lectures before Royal Institution (pub. 1807). The diagram is in fact from his lecture on hydraulics, not light, and although he describes a two-slit experiment, he never claims he actually did it. Furthermore the wavelengths he quotes for the wavelengths of light is from the Newton's rings experiment. There is no clear evidence that Young ever performed the experiment.

Tony Rothman

explain me the double slit diffraction (Radhika thakur (talk) 15:19, 21 April 2012 (UTC))

 * See Double-slit experiment. Also see the comment in this (Diffraction) article about the difference between interference and diffraction. Epzcaw (talk) 07:52, 13 July 2012 (UTC)

Connection to the Heisenberg Uncertainty Principle?
I'm posting this here with the hope that a definitive answer could be included in an amended version of the main article. It was once explained to me in class when I was an undergraduate that single-slit diffraction could be understood in terms of the Heisenberg uncertainty principle. The argument goes as follows: Imagine an incident plane wave of light at wavelength "λ" approaching a 1-dimensional slit aperture from the left, which I will define as the "x" direction. The slit is of height "d" along the "y" direction, and for simplicity's sake is infinite along the z direction. Light must pass through the slit in the form of photons, and the Heisenberg uncertainty principle places limits on these photons' momenta. Specifically, we know that as a photon passes through the slit, its vertical position is known to within an uncertainty "d/2," so the vertical component of momentum, "p_y", is only defined to within a precision of hbar / d.

Now, because the incident light was coming in with wavelength λ, we know that each photon’s total momentum must be 2 π hbar / λ. We can combine the two relationships to give the following relation,
 * $$2 \pi d \sin \theta=\lambda.$$

To within some factors of π and 2 that can probably be explained away by the specific geometry of the slit, this is exactly the width of the central peak in Fraunhofer diffraction.

My question is this: What happens when the slit becomes narrower than lambda / (2 pi)? That is to say, what happens when the slit becomes so narrow that a photon passing though it must acquire an uncertainty in transverse momentum which is greater than the photon’s total momentum was to begin with? Two possibilities come to mind:
 * (a) When the slit gets that narrow, light simply cannot get through. This seems reasonable except for the fact that once you close down the slit even a single photon that sneaks through the slit would have to violate the uncertainty principle. Transmittance that is identically zero seems difficult to swallow.
 * (b) When the slit becomes very narrow, it begins to resemble a cavity where photons can be up-converted to higher energy and momentum. Would this mean that forcing light to pass through a narrow slit changes its color?

Thanks in advance for any thoughts. Csmallw (talk) 04:57, 29 June 2013 (UTC)

Offering a MUCH better example of diffraction - Don Quixote's Windmill.
When you look at "point-like" lights through a rectangular grid, you see four diffraction spikes around each light. This is called a Don Quixote's Windmill, and that should be a heading that re-directs to this page. The best and most common example of this is looking at outside street lighting through an insect screened window or door at night time. Amazingly perceptive yet totally simple spectrographic analysis can be performed just from this phenomenon alone. For example, the difference between incandescent and metal-vapour emission lighting (such as sodium)is immediately obvious. Other examples are looking through lace or sheer curtains. I am happy to take and post some images and do a brief write-up of all of this, but someone else would have to do the redirect. 121.216.26.225 (talk) 09:54, 9 November 2014 (UTC)

 * The phenomenon you describe is indeed a nice everyday example of diffraction, so images of that be welcome. As for its name, a quick search on Google Books didn't turn up a clear reference for "light diffraction Don Quixote". Fgnievinski (talk) 14:11, 9 November 2014 (UTC)

OK, then, will do. My WIkiactivity is very erratice so it will take some time. — Preceding unsigned comment added by 121.216.207.197 (talk) 10:52, 10 November 2014 (UTC)

 * I am bewildered. Iam familiar with this effect but am new to the name. I came looking for an explanation of how a Bahtinov mask works. But Bahtinov masks and lace curtains do not have sizes that approach the wavelengths of the (generally) incoherent light affected, they are just repeating patterns - so how can diffraction occur with patterns at such macro scales? The article needs to explain this as it is a phenomenon the article suggests can't happen. Stub Mandrel (talk) 09:10, 19 May 2015 (UTC)
 * I've removed it from the article for now, given the item you mention is a redlink, we don't have any cite to meet the verifiability policy for it, and considering the concern raised here that it seems to contradict the other cited content. DMacks (talk) 14:12, 19 May 2015 (UTC)

"... Diffraction in the atmosphere by small particles can cause a bright ring to be visible around a bright light source like the sun or the moon. ...
Err, no these bright rings are caused by atmospheric REFRACTION. If you follow the link it even warns you not to confuse the two. 121.216.26.225 (talk) 09:58, 9 November 2014 (UTC)

 * When the particles are small and, accordingly, the mechanism ambiguous, we really ought to use the word "scattering" rather than "diffraction", "refraction", or "reflection". In this case, if the particles are opaque then "refraction" is definitely not the right word for what is happening.2001:480:91:3304:0:0:0:658 (talk) 19:38, 25 June 2021 (UTC)

Diffraction Pattern of red Laser beam
This image on the front page is named as "Laser Interference" by the uploader. Could the author actually upload a interference pattern rather than diffraction pattern? As far as I know, it's possible to produce such pattern with Michelson and Morley setup. — Preceding unsigned comment added by Mywtfmp3 (talk • contribs) 07:04, 8 September 2015 (UTC)

External links modified
Hello fellow Wikipedians,

I have just added archive links to 1 one external link on Diffraction. Please take a moment to review my edit. If necessary, add after the link to keep me from modifying it. Alternatively, you can add to keep me off the page altogether. I made the following changes:
 * Added archive http://web.archive.org/web/20160101000520/http://scripts.mit.edu/~raskar/lightfields/index.php?title=An_Introduction_to_The_Wigner_Distribution_in_Geometric_Optics to http://scripts.mit.edu/~raskar/lightfields/index.php?title=An_Introduction_to_The_Wigner_Distribution_in_Geometric_Optics

When you have finished reviewing my changes, please set the checked parameter below to true to let others know.

Cheers.—cyberbot II  Talk to my owner :Online 08:53, 27 February 2016 (UTC)

title: Talk:Diffraction


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Dynamic Host Configuration Protocol

The Dynamic Host Configuration Protocol (DHCP) is a network management protocol used on Internet Protocol (IP) networks for automatically assigning IP addresses and other communication parameters to devices connected to the network using a client–server architecture.

The technology eliminates the need for individually configuring network devices manually, and consists of two network components, a centrally installed network DHCP server and client instances of the protocol stack on each computer or device. When connected to the network, and periodically thereafter, a client requests a set of parameters from the server using DHCP.

DHCP can be implemented on networks ranging in size from residential networks to large campus networks and regional ISP networks. Many routers and residential gateways have DHCP server capability. Most residential network routers receive a unique IP address within the ISP network. Within a local network, a DHCP server assigns a local IP address to each device.

DHCP services exist for networks running Internet Protocol version 4 (IPv4), as well as version 6 (IPv6). The IPv6 version of the DHCP protocol is commonly called DHCPv6.

History
The Reverse Address Resolution Protocol (RARP) was defined in 1984 for the configuration of simple devices, such as diskless workstations, with a suitable IP address. Acting in the data link layer, it made implementation difficult on many server platforms. It required that a server be present on each individual network link. RARP was superseded by the Bootstrap Protocol (BOOTP) defined in September 1985. This introduced the concept of a relay agent, which allowed the forwarding of BOOTP packets across networks, allowing one central BOOTP server to serve hosts on many IP subnets.

DHCP was first defined in October 1993. It is based on BOOTP, but can dynamically allocate IP addresses from a pool and reclaim them when they are no longer in use. It can also be used to deliver a wide range of extra configuration parameters to IP clients, including platform-specific parameters.

Four years later, the DHCPINFORM message type (used for WPAD) and other small changes were added. This definition, from 1997, remains the core of the standard for IPv4 networks.

DHCPv6 was initially defined in 2003. After updates by many subsequent RFCs, its definition was replaced in 2018, where prefix delegation and stateless address autoconfiguration were now merged.

Overview
Internet Protocol (IP) defines how devices communicate within and across local networks on the Internet. A DHCP server can manage IP settings for devices on its local network, e.g., by assigning IP addresses to those devices automatically and dynamically.

DHCP operates based on the client–server model. When a computer or other device connects to a network, the DHCP client software sends a DHCP broadcast query requesting the necessary information. Any DHCP server on the network may service the request. The DHCP server manages a pool of IP addresses and information about client configuration parameters such as default gateway, domain name, the name servers, and time servers. On receiving a DHCP request, the DHCP server may respond with specific information for each client, as previously configured by an administrator, or with a specific address and any other information valid for the entire network and for the time period for which the allocation (lease) is valid. A DHCP client typically queries this information immediately after booting, and periodically thereafter before the expiration of the information. When a DHCP client refreshes an assignment, it initially requests the same parameter values, but the DHCP server may assign a new address based on the assignment policies set by administrators.

On large networks that consist of multiple links, a single DHCP server may service the entire network when aided by DHCP relay agents located on the interconnecting routers. Such agents relay messages between DHCP clients and DHCP servers located on different subnets.

Depending on implementation, the DHCP server may have three methods of allocating IP addresses:

 * Dynamic allocation: A network administrator reserves a range of IP addresses for DHCP, and each DHCP client on the LAN is configured to request an IP address from the DHCP server during network initialization. The request-and-grant process uses a lease concept with a controllable time period, allowing the DHCP server to reclaim and then reallocate IP addresses that are not renewed.

 * Automatic allocation: The DHCP server permanently assigns an IP address to a requesting client from a range defined by an administrator. This is like dynamic allocation, but the DHCP server keeps a table of past IP address assignments, so that it can preferentially assign to a client the same IP address that the client previously had.

 * Manual allocation: This method is also variously called static DHCP allocation, fixed address allocation, reservation, and MAC/IP address binding. An administrator maps a unique identifier (a client id or MAC address) for each client to an IP address, which is offered to the requesting client. DHCP servers may be configured to fall back to other methods if this fails.

DHCP services are used for Internet Protocol version 4 (IPv4) and IPv6. The details of the protocol for IPv4 and IPv6 differ sufficiently that they may be considered separate protocols. For the IPv6 operation, devices may alternatively use stateless address autoconfiguration. IPv6 hosts may also use link-local addressing to achieve operations restricted to the local network link.

Operation

The DHCP employs a connectionless service model, using the User Datagram Protocol (UDP). It is implemented with two UDP port numbers for its operations which are the same as for the bootstrap protocol (BOOTP). The server listens on UDP port number 67, and the client listens on UDP port number 68.

DHCP operations fall into four phases: server discovery, IP lease offer, IP lease request, and IP lease acknowledgement. These stages are often abbreviated as DORA for discovery, offer, request, and acknowledgement.

The DHCP operation begins with clients broadcasting a request. If the client and server are in different Broadcast Domains, a DHCP Helper or DHCP Relay Agent may be used. Clients requesting renewal of an existing lease may communicate directly via UDP unicast, since the client already has an established IP address at that point. Additionally, there is a BROADCAST flag (1 bit in 2 byte flags field, where all other bits are reserved and so are set to 0) the client can use to indicate in which way (broadcast or unicast) it can receive the DHCPOFFER: 0x8000 for broadcast, 0x0000 for unicast. Usually, the DHCPOFFER is sent through unicast. For those hosts which cannot accept unicast packets before IP addresses are configured, this flag can be used to work around this issue.

Discovery
The DHCP client broadcasts a DHCPDISCOVER message on the network subnet using the destination address (limited broadcast) or the specific subnet broadcast address (directed broadcast). A DHCP client may also request an IP address in the DHCPDISCOVER, which the server may take into account when selecting an address to offer.

For example, if HTYPE is set to 1, to specify that the medium used is Ethernet, HLEN is set to 6 because an Ethernet address (MAC address) is 6 octets long. The CHADDR is set to the MAC address used by the client. Some options are set as well.

Offer
When a DHCP server receives a DHCPDISCOVER message from a client, which is an IP address lease request, the DHCP server reserves an IP address for the client and makes a lease offer by sending a DHCPOFFER message to the client. This message contains the client's client id (traditionally a MAC address), the IP address that the server is offering, the subnet mask, the lease duration, and the IP address of the DHCP server making the offer. The DHCP server may also take notice of the hardware-level MAC address in the underlying transport layer: according to current RFCs the transport layer MAC address may be used if no client ID is provided in the DHCP packet.

The DHCP server determines the configuration based on the client's hardware address as specified in the CHADDR (client hardware address) field. In the following example the server specifies the client's IP address in the YIADDR (your IP address) field.

Request
In response to the DHCP offer, the client replies with a DHCPREQUEST message, broadcast to the server, requesting the offered address. A client can receive DHCP offers from multiple servers, but it will accept only one DHCP offer. Before claiming an IP address, the client will broadcast an ARP request, in order to find if there is another host present in the network with the proposed IP address. If there is no reply, this address does not conflict with that of another host, so it is free to be used.

The client must send the server identification option in the DHCPREQUEST message, indicating the server whose offer the client has selected. When other DHCP servers receive this message, they withdraw any offers that they have made to the client and return their offered IP address to the pool of available addresses.

Acknowledgement
When the DHCP server receives the DHCPREQUEST message from the client, the configuration process enters its final phase. The acknowledgement phase involves sending a DHCPACK packet to the client. This packet includes the lease duration and any other configuration information that the client might have requested. At this point, the IP configuration process is completed.

The protocol expects the DHCP client to configure its network interface with the negotiated parameters.

After the client obtains an IP address, it should probe the newly received address (e.g. with ARP Address Resolution Protocol) to prevent address conflicts caused by overlapping address pools of DHCP servers. If this probe finds another computer using that address, the computer should send DHCPDECLINE, broadcast, to the server.

Information
A DHCP client may request more information than the server sent with the original DHCPOFFER. The client may also request repeat data for a particular application. For example, browsers use DHCP Inform to obtain web proxy settings via WPAD.

Releasing
The client sends a request to the DHCP server to release the DHCP information and the client deactivates its IP address. As client devices usually do not know when users may unplug them from the network, the protocol does not mandate the sending of DHCP Release.

Client configuration parameters
A DHCP server can provide optional configuration parameters to the client. RFC 2132 describes the available DHCP options defined by Internet Assigned Numbers Authority (IANA) - DHCP and BOOTP PARAMETERS.

A DHCP client can select, manipulate and overwrite parameters provided by a DHCP server. In Unix-like systems this client-level refinement typically takes place according to the values in the configuration file /etc/dhclient.conf.

Options
Options are octet strings of varying length. This is called Type–length–value encoding. The first octet is the option code, the second octet is the number of following octets and the remaining octets are code dependent. For example, the DHCP message-type option for an offer would appear as 0x35, 0x01, 0x02, where 0x35 is code 53 for "DHCP message type", 0x01 means one octet follows and 0x02 is the value of "offer".

The following tables list the available DHCP options.

DHCP message types
This table lists the DHCP message types, documented in RFC 2132, RFC 3203, RFC 4388, RFC 6926 and RFC 7724. These codes are the value in the DHCP extension 53, shown in the table above.

Client vendor identification
An option exists to identify the vendor and functionality of a DHCP client. The information is a variable-length string of characters or octets which has a meaning specified by the vendor of the DHCP client. One method by which a DHCP client can communicate to the server that it is using a certain type of hardware or firmware is to set a value in its DHCP requests called the Vendor Class Identifier (VCI) (Option 60).

The value to which this option is set gives the DHCP server a hint about any required extra information that this client needs in a DHCP response. Some types of set-top boxes set the VCI to inform the DHCP server about the hardware type and functionality of the device. An Aruba campus wireless access point, for example, supplies value 'ArubaAP' as option 60 in its DHCPDISCOVER message. The DHCP server can then augment its DHCPOFFER with an IP address of an Aruba wireless controller in option 43, so the access point knows where to register itself.

Setting a VCI by the client allows a DHCP server to differentiate between client machines and process the requests from them appropriately.

Relay agent information sub-options
The relay agent information option (option 82) specifies container for attaching sub-options to DHCP requests transmitted between a DHCP relay and a DHCP server.

Relaying
In small networks, where only one IP subnet is being managed, DHCP clients communicate directly with DHCP servers. However, DHCP servers can also provide IP addresses for multiple subnets. In this case, a DHCP client that has not yet acquired an IP address cannot communicate directly with a DHCP server not on the same subnet, as the client's broadcast can only be received on its own subnet.

In order to allow DHCP clients on subnets not directly served by DHCP servers to communicate with DHCP servers, DHCP relay agents can be installed on these subnets. A DHCP relay agent runs on a network device, capable of routing between the client's subnet and the subnet of the DHCP server. The DHCP client broadcasts on the local link; the relay agent receives the broadcast and transmits it to one or more DHCP servers using unicast. The IP addresses of the DHCP servers are manually configured in the relay agent. The relay agent stores its own IP address, from the interface on which it has received the client's broadcast, in the GIADDR field of the DHCP packet. The DHCP server uses the GIADDR-value to determine the subnet, and subsequently the corresponding address pool, from which to allocate an IP address. When the DHCP server replies to the client, it sends the reply to the GIADDR-address, again using unicast. The relay agent then retransmits the response on the local network, using unicast (in most cases) to the newly reserved IP address, in an Ethernet frame directed to the client's MAC address. The client should accept the packet as its own, even when that IP address is not yet set on the interface. Directly after processing the packet, the client sets the IP address on its interface and is ready for regular IP communication, directly thereafter.

If the client's implementation of the IP stack does not accept unicast packets when it has no IP address yet, the client may set the broadcast bit in the FLAGS field when sending a DHCPDISCOVER packet. The relay agent will use the broadcast IP address (and the clients MAC address) to inform the client of the server's DHCPOFFER.

The communication between the relay agent and the DHCP server typically uses both a source and destination UDP port of 67.

Client states

A DHCP client can receive these messages from a server:
 * DHCPOFFER
 * DHCPACK
 * DHCPNAK

The client moves through DHCP states depending on how the server responds to the messages that the client sends.

Reliability
The DHCP ensures reliability in several ways: periodic renewal, rebinding, and failover. DHCP clients are allocated leases that last for some period of time. Clients begin to attempt to renew their leases once half the lease interval has expired. They do this by sending a unicast DHCPREQUEST message to the DHCP server that granted the original lease. If that server is down or unreachable, it will fail to respond to the DHCPREQUEST. However, in that case the client repeats the DHCPREQUEST from time to time, so if the DHCP server comes back up or becomes reachable again, the DHCP client will succeed in contacting it and renew the lease.

If the DHCP server is unreachable for an extended period of time, the DHCP client will attempt to rebind, by broadcasting its DHCPREQUEST rather than unicasting it. Because it is broadcast, the DHCPREQUEST message will reach all available DHCP servers. If some other DHCP server is able to renew the lease, it will do so at this time.

In order for rebinding to work, when the client successfully contacts a backup DHCP server, that server must have accurate information about the client's binding. Maintaining accurate binding information between two servers is a complicated problem; if both servers are able to update the same lease database, there must be a mechanism to avoid conflicts between updates on the independent servers. A proposal for implementing fault-tolerant DHCP servers was submitted to the Internet Engineering Task Force, but never formalized.

If rebinding fails, the lease will eventually expire. When the lease expires, the client must stop using the IP address granted to it in its lease. At that time it will restart the DHCP process from the beginning by broadcasting a  message. Since its lease has expired, it will accept any IP address offered to it. Once it has a new IP address (presumably from a different DHCP server) it will once again be able to use the network. However, since its IP address has changed, any ongoing connections will be broken.

IPv6 networks
The basic methodology of DHCP was developed for networks based on Internet Protocol version 4 (IPv4). Since the development and deployment of IPv6 networks, DHCP has also been used for assigning parameters in such networks, despite the inherent features of IPv6 for stateless address autoconfiguration. The IPv6 version of the protocol is designated as DHCPv6.

Security
The base DHCP does not include any mechanism for authentication. Because of this, it is vulnerable to a variety of attacks. These attacks fall into three main categories:

 * Unauthorized DHCP servers providing false information to clients.
 * Unauthorized clients gaining access to resources.
 * Resource exhaustion attacks from malicious DHCP clients.

Because the client has no way to validate the identity of a DHCP server, unauthorized DHCP servers (commonly called "rogue DHCP") can be operated on networks, providing incorrect information to DHCP clients. This can serve either as a denial-of-service attack, preventing the client from gaining access to network connectivity, or as a man-in-the-middle attack. Because the DHCP server provides the DHCP client with server IP addresses, such as the IP address of one or more DNS servers, an attacker can convince a DHCP client to do its DNS lookups through its own DNS server, and can therefore provide its own answers to DNS queries from the client. This in turn allows the attacker to redirect network traffic through itself, allowing it to eavesdrop on connections between the client and network servers it contacts, or to simply replace those network servers with its own.

Because the DHCP server has no secure mechanism for authenticating the client, clients can gain unauthorized access to IP addresses by presenting credentials, such as client identifiers, that belong to other DHCP clients. This also allows DHCP clients to exhaust the DHCP server's store of IP addresses—by presenting new credentials each time it asks for an address, the client can consume all the available IP addresses on a particular network link, preventing other DHCP clients from getting service.

DHCP does provide some mechanisms for mitigating these problems. The Relay Agent Information Option protocol extension (usually referred to in the industry by its actual number as Option 82 ) allows network operators to attach tags to DHCP messages as these messages arrive on the network operator's trusted network. This tag is then used as an authorization token to control the client's access to network resources. Because the client has no access to the network upstream of the relay agent, the lack of authentication does not prevent the DHCP server operator from relying on the authorization token.

Another extension, Authentication for DHCP Messages (RFC 3118), provides a mechanism for authenticating DHCP messages. As of 2002, this extension had not seen widespread adoption because of the problems of managing keys for large numbers of DHCP clients. A 2007 book about DSL technologies remarked that:"[T]here were numerous security vulnerabilities identified against the security measures proposed by RFC 3118. This fact, combined with the introduction of 802.1x, slowed the deployment and take-rate of authenticated DHCP, and it has never been widely deployed." A 2010 book notes that:"[T]here have been very few implementations of DHCP Authentication. The challenges of key management and processing delays due to hash computation have been deemed too heavy a price to pay for the perceived benefits."

Architectural proposals from 2008 involve authenticating DHCP requests using 802.1x or PANA (both of which transport EAP). An IETF proposal was made for including EAP in DHCP itself, the so-called EAPoDHCP ; this does not appear to have progressed beyond IETF draft level, the last of which dates to 2010.

IETF standards documents

 * , Dynamic Host Configuration Protocol
 * , DHCP Options and BOOTP Vendor Extensions
 * , DHCP Relay Agent Information Option
 * , Dynamic Host Configuration Protocol (DHCP) Domain Search Option
 * , Reclassifying Dynamic Host Configuration Protocol Version Four (DHCPv4) Options
 * , Information Refresh Time Option for Dynamic Host Configuration Protocol for IPv6
 * , Node-specific Client Identifiers for Dynamic Host Configuration Protocol Version Four (DHCPv4)
 * , Detecting Network Attachment in IPv4 (DNAv4)
 * , Classless Static Route Option for Dynamic Host Configuration Protocol (DHCP) version 4
 * , DHCP reconfigure extension
 * , Dynamic Host Configuration Protocol (DHCP) Leasequery
 * , DHCPv4 Bulk Leasequery
 * , Active DHCPv4 Lease Query

title: Dynamic Host Configuration Protocol


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Dava Sobel

Dava Sobel (born June 15, 1947) is an American writer of popular expositions of scientific topics. Her books include Longitude, about English clockmaker John Harrison; Galileo's Daughter, about Galileo's daughter Maria Celeste; and The Glass Universe: How the Ladies of the Harvard Observatory Took the Measure of the Stars about the Harvard Computers.

Biography
Sobel was born in The Bronx, New York City. She graduated from the Bronx High School of Science and Binghamton University. She wrote Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time in 1995. The story was made into a television movie, of the same name by Charles Sturridge and Granada Film in 1999, and was shown in the United States by A&E.

Her book Galileo's Daughter: A Historical Memoir of Science, Faith, and Love was a finalist for the 2000 Pulitzer Prize for Biography or Autobiography. She holds honorary doctor of letters degrees from the University of Bath and Middlebury College, Vermont, both awarded in 2002.

Sobel made her first foray into teaching at the University of Chicago as the Vare Writer-in-Residence in the winter of 2006. She taught a one-quarter seminar on writing about science.

She served as a judge for the PEN/E. O. Wilson Literary Science Writing Award in 2012.

Sobel is the niece of journalist Ruth Gruber and the cousin of epidemiologist David Michaels.

Legacy
Asteroid 30935 Davasobel, discovered by Carolyn S. Shoemaker and David H. Levy was named after her for her literary work in physics.

Sobel states she is a chaser of solar eclipses and that "it's the closest thing to witnessing a miracle". As of August 2012 she had seen eight, and planned to see the November 2012 total solar eclipse in Australia.

Publications

 * Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time (1995) ISBN 1-85702-571-7. – the genius in question was John Harrison, who spent decades trying to convince the British Admiralty of the accuracy of his naval timepieces and their use in determining longitude when at sea in order to win the longitude prize. The book itself won the 1997 British Book of the Year award.
 * Galileo's Daughter: A Historical Memoir of Science, Faith, and Love (2000) ISBN 0-14-028055-3
 * The Best American Science Writing 2004 (editor) ISBN 9780060726409,
 * The Planets: A discourse on the discovery, science, history and mythology, of the planets in our solar system, with one chapter devoted to each of the celestial spheres. (2005) ISBN 1-85702-850-3,
 * The Glass Universe: How the Ladies of the Harvard Observatory Took the Measure of the Stars (2016) ISBN 9780143111344,
 * The Best American Science Writing 2004 (editor) ISBN 9780060726409,
 * The Planets: A discourse on the discovery, science, history and mythology, of the planets in our solar system, with one chapter devoted to each of the celestial spheres. (2005) ISBN 1-85702-850-3,
 * The Glass Universe: How the Ladies of the Harvard Observatory Took the Measure of the Stars (2016) ISBN 9780143111344,
 * The Glass Universe: How the Ladies of the Harvard Observatory Took the Measure of the Stars (2016) ISBN 9780143111344,

Recognition
She was named a Fellow of the American Physical Society in 2022 "for outstanding writings covering many centuries of key developments in physics and astronomy and the people central to those developments".

title: Dava Sobel


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Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.

Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.

Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics. Outside of physics, differential geometry finds applications in chemistry, economics, engineering, control theory, computer graphics and computer vision, and recently in machine learning.

History and development
The history and development of differential geometry as a subject begins at least as far back as classical antiquity. It is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology, especially the study of manifolds. In this section we focus primarily on the history of the application of infinitesimal methods to geometry, and later to the ideas of tangent spaces, and eventually the development of the modern formalism of the subject in terms of tensors and tensor fields.

Classical antiquity until the Renaissance (300 BC – 1600 AD)
The study of differential geometry, or at least the study of the geometry of smooth shapes, can be traced back at least to classical antiquity. In particular, much was known about the geometry of the Earth, a spherical geometry, in the time of the ancient Greek mathematicians. Famously, Eratosthenes calculated the circumference of the Earth around 200 BC, and around 150 AD Ptolemy in his Geography introduced the stereographic projection for the purposes of mapping the shape of the Earth. Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in geodesy, although in a much simplified form. Namely, as far back as Euclid's Elements it was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to the surface of the Earth leads to the conclusion that great circles, which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Indeed, the measurements of distance along such geodesic paths by Eratosthenes and others can be considered a rudimentary measure of arclength of curves, a concept which did not see a rigorous definition in terms of calculus until the 1600s.

Around this time there were only minimal overt applications of the theory of infinitesimals to the study of geometry, a precursor to the modern calculus-based study of the subject. In Euclid's Elements the notion of tangency of a line to a circle is discussed, and Archimedes applied the method of exhaustion to compute the areas of smooth shapes such as the circle, and the volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders.

There was little development in the theory of differential geometry between antiquity and the beginning of the Renaissance. Before the development of calculus by Newton and Leibniz, the most significant development in the understanding of differential geometry came from Gerardus Mercator's development of the Mercator projection as a way of mapping the Earth. Mercator had an understanding of the advantages and pitfalls of his map design, and in particular was aware of the conformal nature of his projection, as well as the difference between praga, the lines of shortest distance on the Earth, and the directio, the straight line paths on his map. Mercator noted that the praga were oblique curvatur in this projection. This fact reflects the lack of a metric-preserving map of the Earth's surface onto a flat plane, a consequence of the later Theorema Egregium of Gauss.

After calculus (1600–1800)
The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from calculus began around the 1600s when calculus was first developed by Gottfried Leibniz and Isaac Newton. At this time, the recent work of René Descartes introducing analytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time Pierre de Fermat, Newton, and Leibniz began the study of plane curves and the investigation of concepts such as points of inflection and circles of osculation, which aid in the measurement of curvature. Indeed, already in his first paper on the foundations of calculus, Leibniz notes that the infinitesimal condition $$d^2 y = 0$$ indicates the existence of an inflection point. Shortly after this time the Bernoulli brothers, Jacob and Johann made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated by L'Hopital into the first textbook on differential calculus, the tangents to plane curves of various types are computed using the condition $$dy=0$$, and similarly points of inflection are calculated. At this same time the orthogonality between the osculating circles of a plane curve and the tangent directions is realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of curvature, is written down.

In the wake of the development of analytic geometry and plane curves, Alexis Clairaut began the study of space curves at just the age of 16. In his book Clairaut introduced the notion of tangent and subtangent directions to space curves in relation to the directions which lie along a surface on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the tangent space of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology of curvature and double curvature, essentially the notion of principal curvatures later studied by Gauss and others.

Around this same time, Leonhard Euler, originally a student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly. In regards to differential geometry, Euler studied the notion of a geodesic on a surface deriving the first analytical geodesic equation, and later introduced the first set of intrinsic coordinate systems on a surface, beginning the theory of intrinsic geometry upon which modern geometric ideas are based. Around this time Euler's study of mechanics in the Mechanica lead to the realization that a mass traveling along a surface not under the effect of any force would traverse a geodesic path, an early precursor to the important foundational ideas of Einstein's general relativity, and also to the Euler–Lagrange equations and the first theory of the calculus of variations, which underpins in modern differential geometry many techniques in symplectic geometry and geometric analysis. This theory was used by Lagrange, a co-developer of the calculus of variations, to derive the first differential equation describing a minimal surface in terms of the Euler–Lagrange equation. In 1760 Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem.

Later in the 1700s, the new French school led by Gaspard Monge began to make contributions to differential geometry. Monge made important contributions to the theory of plane curves, surfaces, and studied surfaces of revolution and envelopes of plane curves and space curves. Several students of Monge made contributions to this same theory, and for example Charles Dupin provided a new interpretation of Euler's theorem in terms of the principle curvatures, which is the modern form of the equation.

Intrinsic geometry and non-Euclidean geometry (1800–1900)
The field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann, and also in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non-Euclidean geometry and throughout the same period the development of projective geometry.

Dubbed the single most important work in the history of differential geometry, in 1827 Gauss produced the Disquisitiones generales circa superficies curvas detailing the general theory of curved surfaces. In this work and his subsequent papers and unpublished notes on the theory of surfaces, Gauss has been dubbed the inventor of non-Euclidean geometry and the inventor of intrinsic differential geometry. In his fundamental paper Gauss introduced the Gauss map, Gaussian curvature, first and second fundamental forms, proved the Theorema Egregium showing the intrinsic nature of the Gaussian curvature, and studied geodesics, computing the area of a geodesic triangle in various non-Euclidean geometries on surfaces.

At this time Gauss was already of the opinion that the standard paradigm of Euclidean geometry should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles. Around this same time János Bolyai and Lobachevsky independently discovered hyperbolic geometry and thus demonstrated the existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced by Eugenio Beltrami later in the 1860s, and Felix Klein coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing. Implicitly, the spherical geometry of the Earth that had been studied since antiquity was a non-Euclidean geometry, an elliptic geometry.

The development of intrinsic differential geometry in the language of Gauss was spurred on by his student, Bernhard Riemann in his Habilitationsschrift, On the hypotheses which lie at the foundation of geometry. In this work Riemann introduced the notion of a Riemannian metric and the Riemannian curvature tensor for the first time, and began the systematic study of differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by $$ds^2$$ by Riemann, was the development of an idea of Gauss's about the linear element $$ds$$ of a surface. At this time Riemann began to introduce the systematic use of linear algebra and multilinear algebra into the subject, making great use of the theory of quadratic forms in his investigation of metrics and curvature. At this time Riemann did not yet develop the modern notion of a manifold, as even the notion of a topological space had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of spacetime through the analysis of masses within spacetime, linking with the earlier observation of Euler that masses under the effect of no forces would travel along geodesics on surfaces, and predicting Einstein's fundamental observation of the equivalence principle a full 60 years before it appeared in the scientific literature.

In the wake of Riemann's new description, the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of the study of curves and surfaces to a more systematic approach in terms of tensor calculus and Klein's Erlangen program, and progress increased in the field. The notion of groups of transformations was developed by Sophus Lie and Jean Gaston Darboux, leading to important results in the theory of Lie groups and symplectic geometry. The notion of differential calculus on curved spaces was studied by Elwin Christoffel, who introduced the Christoffel symbols which describe the covariant derivative in 1868, and by others including Eugenio Beltrami who studied many analytic questions on manifolds. In 1899 Luigi Bianchi produced his Lectures on differential geometry which studied differential geometry from Riemann's perspective, and a year later Tullio Levi-Civita and Gregorio Ricci-Curbastro produced their textbook systematically developing the theory of absolute differential calculus and tensor calculus. It was in this language that differential geometry was used by Einstein in the development of general relativity and pseudo-Riemannian geometry.

Modern differential geometry (1900–2000)
The subject of modern differential geometry emerged from the early 1900s in response to the foundational contributions of many mathematicians, including importantly the work of Henri Poincaré on the foundations of topology. At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as Hilbert's program. As part of this broader movement, the notion of a topological space was distilled in by Felix Hausdorff in 1914, and by 1942 there were many different notions of manifold of a combinatorial and differential-geometric nature.

Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation $$g$$ for a Riemannian metric, and $$\Gamma$$ for the Christoffel symbols, both coming from G in Gravitation. Élie Cartan helped reformulate the foundations of the differential geometry of smooth manifolds in terms of exterior calculus and the theory of moving frames, leading in the world of physics to Einstein–Cartan theory.

Following this early development, many mathematicians contributed to the development of the modern theory, including Jean-Louis Koszul who introduced connections on vector bundles, Shiing-Shen Chern who introduced characteristic classes to the subject and began the study of complex manifolds, Sir William Vallance Douglas Hodge and Georges de Rham who expanded understanding of differential forms, Charles Ehresmann who introduced the theory of fibre bundles and Ehresmann connections, and others. Of particular importance was Hermann Weyl who made important contributions to the foundations of general relativity, introduced the Weyl tensor providing insight into conformal geometry, and first defined the notion of a gauge leading to the development of gauge theory in physics and mathematics.

In the middle and late 20th century differential geometry as a subject expanded in scope and developed links to other areas of mathematics and physics. The development of gauge theory and Yang–Mills theory in physics brought bundles and connections into focus, leading to developments in gauge theory. Many analytical results were investigated including the proof of the Atiyah–Singer index theorem. The development of complex geometry was spurred on by parallel results in algebraic geometry, and results in the geometry and global analysis of complex manifolds were proven by Shing-Tung Yau and others. In the latter half of the 20th century new analytic techniques were developed in regards to curvature flows such as the Ricci flow, which culminated in Grigori Perelman's proof of the Poincaré conjecture. During this same period primarily due to the influence of Michael Atiyah, new links between theoretical physics and differential geometry were formed. Techniques from the study of the Yang–Mills equations and gauge theory were used by mathematicians to develop new invariants of smooth manifolds. Physicists such as Edward Witten, the only physicist to be awarded a Fields medal, made new impacts in mathematics by using topological quantum field theory and string theory to make predictions and provide frameworks for new rigorous mathematics, which has resulted for example in the conjectural mirror symmetry and the Seiberg–Witten invariants.

Riemannian geometry
Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, though they still resemble Euclidean space at each point infinitesimally, i.e. in the first order of approximation. Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all possess natural analogues in Riemannian geometry. The notion of a directional derivative of a function from multivariable calculus is extended to the notion of a covariant derivative of a tensor. Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry.

Pseudo-Riemannian geometry
Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite. A special case of this is a Lorentzian manifold, which is the mathematical basis of Einstein's general relativity theory of gravity.

Finsler geometry
Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric, that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold $$M$$ is a function $$F:\mathrm{T}M\to[0,\infty)$$ such that:
 * 1) $$F(x,my)=mF(x,y)$$ for all $$x,y$$ in $$\mathrm{T}M$$ and all $$m\ge 0$$,
 * 2) $$F$$ is infinitely differentiable in $$\mathrm{T}M\setminus\{0\}$$,
 * 3) The vertical Hessian of $$F^2$$ is positive definite.

Symplectic geometry
Symplectic geometry is the study of symplectic manifolds. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e., a nondegenerate 2-form ω, called the symplectic form. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: dω = 0.

A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Joseph Louis Lagrange on analytical mechanics and later in Carl Gustav Jacobi's and William Rowan Hamilton's formulations of classical mechanics.

By contrast with Riemannian geometry, where the curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré–Birkhoff theorem, conjectured by Henri Poincaré and then proved by G.D. Birkhoff in 1912. It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then the map has at least two fixed points.

Contact geometry
Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A contact structure on a (2n + 1)-dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with the level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point p, a hyperplane distribution is determined by a nowhere vanishing 1-form $$\alpha$$, which is unique up to multiplication by a nowhere vanishing function:

 * $$ H_p = \ker\alpha_p\subset T_{p}M.$$

A local 1-form on M is a contact form if the restriction of its exterior derivative to H is a non-degenerate two-form and thus induces a symplectic structure on Hp at each point. If the distribution H can be defined by a global one-form $$\alpha$$ then this form is contact if and only if the top-dimensional form

 * $$\alpha\wedge (d\alpha)^n$$

is a volume form on M, i.e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.

Complex and Kähler geometry
Complex differential geometry is the study of complex manifolds. An almost complex manifold is a real manifold $$M$$, endowed with a tensor of type (1, 1), i.e. a vector bundle endomorphism (called an almost complex structure)
 * $$ J:TM\rightarrow TM $$, such that $$J^2=-1. \,$$

It follows from this definition that an almost complex manifold is even-dimensional.

An almost complex manifold is called complex if $$N_J=0$$, where $$N_J$$ is a tensor of type (2, 1) related to $$J$$, called the Nijenhuis tensor (or sometimes the torsion). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. An almost Hermitian structure is given by an almost complex structure J, along with a Riemannian metric g, satisfying the compatibility condition
 * $$g(JX,JY)=g(X,Y). \,$$

An almost Hermitian structure defines naturally a differential two-form
 * $$\omega_{J,g}(X,Y):=g(JX,Y). \,$$

The following two conditions are equivalent:

 * 1) $$ N_J=0\mbox{ and }d\omega=0 \,$$
 * 2) $$\nabla J=0 \,$$

where $$\nabla$$ is the Levi-Civita connection of $$g$$. In this case, $$(J, g)$$ is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.

CR geometry
CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds.

Conformal geometry
Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

Differential topology
Differential topology is the study of global geometric invariants without a metric or symplectic form.

Differential topology starts from the natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms. Beside Lie algebroids, also Courant algebroids start playing a more important role.

Lie groups
A Lie group is a group in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which is the tangent space at the unit endowed with the Lie bracket between left-invariant vector fields. Beside the structure theory there is also the wide field of representation theory.

Geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology.

Gauge theory
Gauge theory is the study of connections on vector bundles and principal bundles, and arises out of problems in mathematical physics and physical gauge theories which underpin the standard model of particle physics. Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric moduli spaces of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as the Euler–Lagrange equations describing the equations of motion of certain physical systems in quantum field theory, and so their study is of considerable interest in physics.

Bundles and connections
The apparatus of vector bundles, principal bundles, and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of parallel transport. An important example is provided by affine connections. For a surface in R3, tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has a well-known standard definition of metric and parallelism. In Riemannian geometry, the Levi-Civita connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be spacetime and the bundles and connections are related to various physical fields.

Intrinsic versus extrinsic
From the beginning and through the middle of the 19th century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium, to the effect that Gaussian curvature is an intrinsic invariant.

The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic. However, there is a price to pay in technical complexity: the intrinsic definitions of curvature and connections become much less visually intuitive.

These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem.) In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the shape operator.

Applications
Below are some examples of how differential geometry is applied to other fields of science and mathematics.
 * In physics, differential geometry has many applications, including:
 * Differential geometry is the language in which Albert Einstein's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature of spacetime. Understanding this curvature is essential for the positioning of satellites into orbit around the Earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes.
 * Differential forms are used in the study of electromagnetism.
 * Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.
 * Riemannian geometry and contact geometry have been used to construct the formalism of geometrothermodynamics which has found applications in classical equilibrium thermodynamics.
 * In chemistry and biophysics when modelling cell membrane structure under varying pressure.
 * In economics, differential geometry has applications to the field of econometrics.
 * Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry.
 * In engineering, differential geometry can be applied to solve problems in digital signal processing.
 * In control theory, differential geometry can be used to analyze nonlinear controllers, particularly geometric control
 * In probability, statistics, and information theory, one can interpret various structures as Riemannian manifolds, which yields the field of information geometry, particularly via the Fisher information metric.
 * In structural geology, differential geometry is used to analyze and describe geologic structures.
 * In computer vision, differential geometry is used to analyze shapes.
 * In image processing, differential geometry is used to process and analyse data on non-flat surfaces.
 * Grigori Perelman's proof of the Poincaré conjecture using the techniques of Ricci flows demonstrated the power of the differential-geometric approach to questions in topology and it highlighted the important role played by its analytic methods.
 * In wireless communications, Grassmannian manifolds are used for beamforming techniques in multiple antenna systems.
 * In geodesy, for calculating distances and angles on the mean sea level surface of the Earth, modelled by an ellipsoid of revolution.

title: Differential geometry