what is a field in mathmatics?
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alright, lots of people talked about lots of things without any formal definitions.  I am going to assume that you are talking about fields in the algebraic sense.  All the other fields mentioned (vector fields) are specific incidents of this general definition.  To begin we need a set (of numbers).  Traditionally this is a set of numbers such as the integers, rational numbers, real numbers and so on.  On this set we define TWO opperations.  We are going to call the + and * but they do not have to be the standard definitions of addition and multiplication.  For example we could define a+b=2*a+b where * and + on the right stand for the opperations that we are used to.  Alright, so we have a set of numbers that we are going to call F and we have two opperations +,*.  We denote the entire family F,+,*\nF,+,* is a field if all of the following hold:\n1)F,+ is an abelian group\n2)F-{0},* is an abelian group\n3)Distributive law: a*(b+c)=a*b+a*c\nNow that does not sound easy does it?  To understand that we need to know what a group is and what abelian means.  Fist, all that abelian means is that the group is communitive.  Meaning a+b=b+a and a*b=b*a.  There are some sets with specific operations such that they are not communative (ex. matrix multiplication).\nWhat is a group?\nG,+ is a group if the following all hold:\n1)Closure: for all a,b in G, a+b is in G\n2)Associativity: for all a,b,c in G, (a+b)+c=a+(b+c)\n3)Indentity: there is an element "e" in G such that a+e=e+a=a for all a in G\n4)Inverse: for all a in G, there exits a b in G such that a+b=e\n\nThese four properties define a group over the operation +\nTo apply this to our definition of a field, we can see that for F,+,* to be a field, F,+ must be a group. Replace G by F in our definition of a group. Also, F-{0},* must be a group so replace G with F-{0} in the above definition.\n\nSome examples of Fields.  The rational numbers with the standard addition and multiplication.  Are the integers a field?  NO because there is no multiplicative inverse for any of the integers except 1.  This means that Z (the integers) under multiplication is not a group thus it cannot be a field.\n\nAdditional Info:\n  The guy below me claims that a field under multiplication only needs to be a semi-group (closed and associative).  That is not correct.  I belive that he is thinking of a Ring.  A Ring is the same as a field except it only needs to be a semi-group under multiplication.  A field needs to be a group under both operations.
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Lots of fields are rectangular.  The baseball infield is a square.
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The field of electronics is the study and use of systems that operate by controlling the flow of electrons or other electrically charged particles in devices such as thermionic valves and semiconductors. The design and construction of electronic circuits to solve practical problems is part of the fields of electronics engineering, and the hardware design side of computer engineering. The study of new semiconductor devices and their technology is sometimes considered as a branch of physics.
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It depends on what you are talking about, there are many different types of models.\n\nAbstractions, concepts or software\n    * Model (abstract), a conceptual or theoretical representation of a phenomenon\n    o Mathematical model\n    * Model (computer games), in 3D computer graphics, the computer generated image of a character or an object\n    * Computer model, a computer program which attempts to simulate an abstract model of a particular system\n    * Mental model, a person's cognitive representation of an idea or thought process\n    * Model theory, study of the representation of mathematical concepts\n    * Modeling (NLP), a process in neuro-linguistic programming\n    * Similitude (model), in engineering, used in the scientific testing of physical models\n    * Solid modelling, study of unambiguous representations of the solid parts of an object\n    * The Standard Model, the theory in particle physics which describes certain fundamental forces and particles\n    * model building (particle physics)\n\nPhysical objects\n    * Model (physical), a physical representation of an object\n    * Scale model, a replica or prototype of an object\n\nPersons or occupations\n\n    * Model (art), a person who poses for purposes of art, for example in art school (often known as an artist's model)\n    * Model (person), a person whose occupation is to display products, eg. a fashion model\n          o Supermodel, a model who attains celebrity status\n    * Promotional model, a person who promotes a product or service\n    * Role model, a person who serves as a behavioural or moral example to others
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Mathematics- The science of numbers in all their relations and applications. (Math is a general concept including arithmetic, statistics and probability, trig., calculus, and more)\n\nArithmetic- The most elementary branch of mathematics (addition, subtraction, multiplication, division for example)
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"Maths" is a contraction of "mathematics", a short way to say and write it.  It includes all branches of maths, e.g. arithmetic, algebra, geometry, trigonometry, statics, differential and integral calculus, quantum mechanics, etc. As you can see, arithmetic is a subset of mathematics.  Arithmetic deals with the manipulation of real numbers only.
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MATHEMATICS\n\n(MAT HE MAT ICS)
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generally, validation limits what can go in a field.  \n\nfor example, if the field was "email address," you would want to write a validation algorithm that confirmed someone entered a properly formatted email address (ie: doofus@nowhere.com)\n\nas a programmer, YOU have to decide whether a field should be validated.  as you develop your project, ask yourself "is the user smart enough where i don't have to code for it?"  if you're already asking the question, the answer is usually "no."
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Nodes for important fields, attributes for subfields.
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Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics.\n\nWe consider vector fields, which associate a vector to every point in space, and scalar fields, which associate a scalar to every point in space. For example, the temperature of a swimming pool is a scalar field: to each point we associate a scalar value of temperature. The water flow in the same pool is a vector field: to each point we associate a velocity vector.
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In two charged plates, other than between the plates (and the edge effects), are there any field line? \nYES! other field lines are between the outer sides of the plates(depending on the charges of the plates and the convention used for the direction of the flow). To prove this, just connect the outer sides of the plates by a wire through an amper meter and you will see the meter moving.\n\nWhy is the electric field constant everywhere in the space between parallel plates? \nThey are not unless the area of both plates is inifinte and that the two plates are parallel.  In this case, the 1) direction of the field is the same every where since by definition the plates are parallel and 2) the absolute value is constant.  If not, there will be a surface current and the current would make the entire electric field (its magnitude) to be constant over the entire surface.
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Pure mathematics (theorems, definitions and proofs) is developed before the applications; e.g., why the number of processors is relatively prime to the number of memory locations\n\nAlgorithms are developed in a hope to improve something that already exists\n\nThe EM algorithm is used in computer vision as well as probability models like data clustering in machine learning \n\nEM stands for Expectation-Maximization where the likelihood of estimates is maximized\n\nThe study of Probability and Statistics involves learning and using many algorithms, including the EM algorithm
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I would say fields such as health specialties, business, and computer science...
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umm none that im aware of, what kind of mathamatics leval are you looking at?
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Which definition of a semi-field are you using?  There seem to be several.
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The field of Medicine.
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The above answers are good guidelines but the ACTUAL mathematic formula is considered patented proprietary information, and nobody knows exactly how it works except for the agencies that use them.  That's why all the information out there is so general....because nobody really knows and Fair Isaac isn't telling.