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Multivariate calculusVector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics.
We 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.
Three operations are important in vector calculus:
- gradient: measures the rate and direction of change in a scalar field; the gradient of a scalar field is a vector field.
- curl: measures a vector field's tendency to rotate about a point; the curl of a vector field is another vector field.
- divergence: measures a vector field's tendency to originate from or converge upon certain points; the divergence of a vector field is a scalar field.
- Stokes' theorem
- divergence theoremMultivariate calculus Theorems In vector calculus, the divergence theorem also known as Gauss' theorem or Ostrogradsky-Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field
Most of the analytic results are more easily understood using the machinery of differential geometry, for which vector calculus forms a subset.
History
QuaternionQuaternion is also a musical composition by Sofia Gubaidulina In mathematics, the quaternions are a non-commutative extension of the complex numbers. They were first described by William Rowan Hamilton of Ireland in 1843. At first, the quaternions were res were discovered by William Rowan HamiltonSir William Rowan Hamilton ( August 4, 1805 September 2, 1865) was an Irish mathematician, physicist, and astronomer. Hamilton's discovery of quaternions is his best known investigation. Hamilton also contributed to the development of optics, dynamics, an of IrelandThe island of Ireland ire in Irish, Airlann in Ulster Scots) is the third-largest island in Europe. It lies on the west side of the Irish Sea, close to the island of Great Britain. It is composed of the Republic of Ireland in the south and Northern Irelan in 1843Events February 6 The first minstrel show in the United States The Virginia Minstrels opens (Bowery Amphitheatre in New York City). February 11 Giuseppe Verdi's opera I Lombardi premieres in Milan May 18 The Disruption of the Church of Scotland took place. Hamilton was looking for ways of extending complex numberThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , whers (which can be viewed as pointThe word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point ins on a plane) to higher spatial dimensions. Quaternions are made of a 3 dimensional vector plus a scalar.
Oliver Heaviside and Willard Gibbs among others developed vector algebra and vector calculus.
Some of Hamilton's supporters vociferously opposed the growing fields of vector algebra and vector calculus (developed by Oliver Heaviside and Willard Gibbs among others), maintaining that quaternions provided a superior notation. While this is debatable in three dimensions, quaternions cannot be used in other dimensions (though extensions like octonions and Clifford algebras may be more applicable). Vector notation nearly universally replaced quaternions in science and engineering by the mid-20th century.
See also: list of multivariable calculus topics.
