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Inhomogeneous Equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric. definition and example of abbreviation More generally, these terms may apply to an or ; the right-hand side is ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operation (mathematics)
In mathematics, an operation is a function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations of finite arity are called finitary operations. A partial operation is defined similarly to an operation, but with a partial function in place of a function. Types of operation There are two common types of operations: unary and binar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infix Notation
Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—" infixed operators"—such as the plus sign in . Usage Binary relations are often denoted by an infix symbol such as set membership ''a'' ∈ ''A'' when the set ''A'' has ''a'' for an element. In geometry, perpendicular lines ''a'' and ''b'' are denoted a \perp b \ , and in projective geometry two points ''b'' and ''c'' are in perspective when b \ \doublebarwedge \ c while they are connected by a projectivity when b \ \barwedge \ c . Infix notation is more difficult to parse by computers than prefix notation (e.g. + 2 2) or postfix notation (e.g. 2 2 +). However many programming languages use it due to its familiarity. It is more used in arithmetic, e.g. 5 × 6. Further notations Infix notation may also be distinguished from function notation, where the name of a function suggests a particular operation, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Space
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a ''perfect'' vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is considerably lower than atmospheric pressure. The Latin term ''in vacuo'' is used to describe an object that is surrounded by a vacuum. The ''quality'' of a partial vacuum refers to how closely it approaches a perfect vacuum. Other things equal, lower gas pressure means higher-quality vacuum. For example, a typical vacuum cleaner produces enough suction to reduce air pressure by around 20%. But high ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Equation
In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; I^1 (u), I^2(u), I^3(u), ..., I^m(u)) = 0where I^i(u) is an integral operator acting on ''u.'' Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; D^1 (u), D^2(u), D^3(u), ..., D^m(u)) = 0where D^i(u) may be viewed as a differential operator of order ''i''. Due to this close connection between differential and integral equations, one can often convert between the two. For examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Simultaneous Equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, :\begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple :(x,y,z)=(1,-2,-2), since it makes all three equations valid. The word "system" indicates that the equations are to be considered collectively, rather than individually. In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A syst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expression (mathematics)
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax. Many authors distinguish an expression from a '' formula'', the former denoting a mathematical object, and the latter denoting a statement about mathematical objects. For example, 8x-5 is an expression, while 8x-5 \geq 5x-8 is a formula. However, in modern mathematics, and in particular in computer algebra, formulas are viewed as expressions that can be evaluated to ''true'' or ''false'', depending on the values that are given to the variables occurring in the expressions. For example 8x-5 \geq 5x-8 takes the value ''false'' if is given a value less than –1, and the value ''true'' otherwise. Examples The use of e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Test Operator
In computer science, a relational operator is a programming language construct or operator that tests or defines some kind of relation between two entities. These include numerical equality (''e.g.'', ) and inequalities (''e.g.'', ). In programming languages that include a distinct boolean data type in their type system, like Pascal, Ada, or Java, these operators usually evaluate to true or false, depending on if the conditional relationship between the two operands holds or not. In languages such as C, relational operators return the integers 0 or 1, where 0 stands for false and any non-zero value stands for true. An expression created using a relational operator forms what is termed a ''relational expression'' or a ''condition''. Relational operators can be seen as special cases of logical predicates. Equality Usage Equality is used in many programming language constructs and data types. It is used to test if an element already exists in a set, or to access to a value th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
