Vector calculus

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Vector calculus

Vector calculus (or vector analysis) is a branch of mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
concerned with differentiation

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
and integration

Integral

Integration is an important concept in mathematics, specifically in the field of calculus and, more broadly, mathematical analysis. Given a function ƒ of a Real number variable x and an interval [a, b] of the real line, the integral...
of vector field

Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space.Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic field or gravity for...
s. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus

Multivariable calculus

Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable....
, which includes vector calculus as well as partial differentiation

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant ....
and multiple integration

Multiple integral

The multiple integral is a type of definite integral extended to Function of more than one real variable, for example, f or f....
. Vector calculus plays an important role in differential geometry and in the study of partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a Relation involving an unknown Function of several independent variables and its partial derivatives with respect to those variables....
s. It is used extensively in physics

Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, especially in the description of electromagnetic

Electromagnetic field

The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
and gravitational field

Gravitational field

A gravitational field is a scientific model used within physics to explain how gravitation exists in the universe. In its original concept, gravity was a force between point masses....
s.

Vector calculus was developed from quaternion

Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space....
analysis by J.

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Encyclopedia

Vector calculus (or vector analysis) is a branch of mathematics

Mathematics

Derivative

Integral

Vector field

Multivariable calculus

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant ....
and multiple integration

Multiple integral

Partial differential equation

Physics

Electromagnetic field

The electromagnetic field is a physical field produced by electric charge. It affects the behavior of charged objects in the vicinity of the field....
and gravitational field

Gravitational field

Quaternion

Oliver Heaviside

Oliver Heaviside was a autodidact English electrical engineering, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's equations in terms of electric and magnetic forces and flux, and independently co-f...
near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Wilson

Edwin Bidwell Wilson

Edwin Bidwell Wilson was an American mathematician and polymath. He was the sole proteg? of Yale's physicist Josiah Willard Gibbs and was mentor to Harvard economist Paul Samuelson....
in their 1901 book, Vector Analysis.

Vector operations

Vector calculus studies various differential operator

Differential operator

In mathematics, a differential operator is an operator defined as a function of the derivative operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another ....
s defined on scalar or vector fields, which are typically expressed in terms of the del

Del

In vector calculus, del is a vector differential operator represented by the nabla symbol: .Del is a mathematical tool serving primarily as a Convention for mathematical notation; it makes many equations easier to comprehend, write, and remember....
operator . The four most important operations in vector calculus are:

Operation	Description	Domain/Range
Gradient Gradient In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....	Measures the rate and direction of change in a scalar field.	Maps scalar fields to vector fields.
Curl	Measures the tendency to rotate about a point in a vector field.	Maps vector fields to vector fields.
Divergence Divergence In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar....	Measures the magnitude of a source or sink at a given point in a vector field.	Maps vector fields to scalar fields.
Laplacian Laplace operator In mathematics and physics, the Laplace operator or Laplacian, denoted by or and named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications....	A composition of the divergence and gradient operations.	Maps scalar fields to scalar fields.

A quantity called the Jacobian

Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.In algebraic geometry the Jacobian of a algebraic curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded....
is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration.

Theorems

Likewise, there are several important theorems related to these operators which generalize the fundamental theorem of calculus

Fundamental theorem of calculus

The fundamental theorem of calculus specifies the relationship between the two central operations of calculus: derivative and integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an antiderivative can be reversed by a differentiation....
to higher dimensions:

Theorem	Statement	Description
Gradient theorem Gradient theorem The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve:...		The line integral Line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use.... through a gradient (vector) field equals the difference in its scalar field at the endpoints of the curve Curve In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum .... .
Green's theorem Green's theorem In physics and mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C....		The integral of the scalar curl of a vector field over some region in the plane equals the line integral of the vector field over the curve bounding the region.
Stokes' theorem Stokes' theorem In differential geometry, Stokes' theorem is a statement about the integral of differential forms which generalizes several theorems from vector calculus....		The integral of the curl of a vector field over a surface Surface In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3.... equals the line integral of the vector field over the curve bounding the surface.
Divergence theorem Divergence theorem In vector calculus, the divergence theorem, also known as Gauss?s theorem , Ostrogradsky?s theorem , or Gauss-Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface....		The integral of the divergence of a vector field over some solid equals the integral of the flux Flux In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.In the study of transport phenomena , flux is defined as the amount that flows through a unit area per unit time*.... through the surface bounding the solid.

The use of vector calculus may require the handedness of the coordinate system

Coordinate system

In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalar to each Point in an n-dimensional space....
to be taken into account (see cross product and handedness

Cross product

In mathematics, the cross product is a binary operation on two vector s in a three-dimensional Euclidean space that results in another vector which is orthogonal to the plane containing the two input vectors....
for more detail). Most of the analytic results are easily understood, in a more general form, using the machinery of differential geometry, of which vector calculus forms a subset.

External links

A Text-book for the Use of Students of Mathematics and Physics, (based upon the lectures of Willard Gibbs) by Edwin Bidwell Wilson

Edwin Bidwell Wilson

Vector calculus

Vector operations

Theorems

See also

External links