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Part of a series of articles about  
Calculus  





Specialized 

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields and fluid flow.
Vector calculus was developed from quaternion analysis by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, Vector Analysis. In the conventional form using cross products, vector calculus does not generalize to higher dimensions, while the alternative approach of geometric algebra, which uses exterior products does generalize, as discussed below.
A scalar field associates a scalar value to every point in a space. The scalar is a mathematical number representing a physical quantity. Examples of scalar fields in applications include the temperature distribution throughout space, the pressure distribution in a fluid, and spinzero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.
A vector field is an assignment of a vector to each point in a space.^{[1]} A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used 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 or gravitational force, as it changes from point to point. This can be used, for example, to calculate work done over a line.
In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientationreversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in geometric algebra, as described below.
The algebraic (nondifferential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field. The basic algebraic operations consist of:
Operation  Notation  Description 

Vector addition  Addition of two vectors, yielding a vector.  
Scalar multiplication  Multiplication of a scalar and a vector, yielding a vector.  
Dot product  Multiplication of two vectors, yielding a scalar.  
Cross product  Multiplication of two vectors in , yielding a (pseudo)vector. 
Also commonly used are the two triple products:
Operation  Notation  Description 

Scalar triple product  The dot product of a vector and a cross product of two vectors.  
Vector triple product  The cross product of a vector and a cross product of two vectors. 
Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator (), also known as "nabla". The three basic vector operators are:
Operation  Notation  Description  Notational analogy 
Domain/Range 

Gradient  Measures the rate and direction of change in a scalar field.  Scalar multiplication  Maps scalar fields to vector fields.  
Divergence  Measures the scalar of a source or sink at a given point in a vector field.  Dot product  Maps vector fields to scalar fields.  
Curl  Measures the tendency to rotate about a point in a vector field in .  Cross product  Maps vector fields to (pseudo)vector fields.  
denotes a scalar field and denotes a vector field 
Also commonly used are the two Laplace operators:
Operation  Notation  Description  Domain/Range 

Laplacian  Measures the difference between the value of the scalar field with its average on infinitesimal balls.  Maps between scalar fields.  
Vector Laplacian  Measures the difference between the value of the vector field with its average on infinitesimal balls.  Maps between vector fields.  
denotes a scalar field and denotes a vector field 
A quantity called the Jacobian matrix is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration.
The three basic vector operators have corresponding theorems which generalize the fundamental theorem of calculus to higher dimensions:
Theorem  Statement  Description  

Gradient theorem  The line integral of the gradient of a scalar field over a curve L is equal to the change in the scalar field between the endpoints p and q of the curve.  
Divergence theorem  The integral of the divergence of a vector field over an ndimensional solid V is equal to the flux of the vector field through the (n−1)dimensional closed boundary surface of the solid.  
Curl (Kelvin–Stokes) theorem  The integral of the curl of a vector field over a surface Σ in is equal to the circulation of the vector field around the closed curve bounding the surface.  
denotes a scalar field and denotes a vector field 
In two dimensions, the divergence and curl theorems reduce to the Green's theorem:
Theorem  Statement  Description  

Green's theorem  The integral of the divergence (or curl) of a vector field over some region A in equals the flux (or circulation) of the vector field over the closed curve bounding the region.  
For divergence, . For curl, . L and M are functions of (x, y). 
Linear approximations are used to replace complicated functions with linear functions that are almost the same. Given a differentiable function with real values, one can approximate for close to by the formula
The righthand side is the equation of the plane tangent to the graph of at
For a continuously differentiable function of several real variables, a point P (that is a set of values for the input variables, which is viewed as a point in R^{n}) is critical if all of the partial derivatives of the function are zero at P, or, equivalently, if its gradient is zero. The critical values are the values of the function at the critical points.
If the function is smooth, or, at least twice continuously differentiable, a critical point may be either a local maximum, a local minimum or a saddle point. The different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.
By Fermat's theorem, all local maxima and minima of a differentiable function occur at critical points. Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.
Vector calculus is particularly useful in studying:
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Vector calculus is initially defined for Euclidean 3space, which has additional structure beyond simply being a 3dimensional real vector space, namely: a norm (giving a notion of length) defined via an inner product (the dot product), which in turn gives a notion of angle, and an orientation, which gives a notion of lefthanded and righthanded. These structures give rise to a volume form, and also the cross product, which is used pervasively in vector calculus.
The gradient and divergence require only the inner product, while the curl and the cross product also requires the handedness of the coordinate system to be taken into account (see cross product and handedness for more detail).
Vector calculus can be defined on other 3dimensional real vector spaces if they have an inner product (or more generally a symmetric nondegenerate form) and an orientation; note that this is less data than an isomorphism to Euclidean space, as it does not require a set of coordinates (a frame of reference), which reflects the fact that vector calculus is invariant under rotations (the special orthogonal group SO(3)).
More generally, vector calculus can be defined on any 3dimensional oriented Riemannian manifold, or more generally pseudoRiemannian manifold. This structure simply means that the tangent space at each point has an inner product (more generally, a symmetric nondegenerate form) and an orientation, or more globally that there is a symmetric nondegenerate metric tensor and an orientation, and works because vector calculus is defined in terms of tangent vectors at each point.
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. Grad and div generalize immediately to other dimensions, as do the gradient theorem, divergence theorem, and Laplacian (yielding harmonic analysis), while curl and cross product do not generalize as directly.
From a general point of view, the various fields in (3dimensional) vector calculus are uniformly seen as being kvector fields: scalar fields are 0vector fields, vector fields are 1vector fields, pseudovector fields are 2vector fields, and pseudoscalar fields are 3vector fields. In higher dimensions there are additional types of fields (scalar/vector/pseudovector/pseudoscalar corresponding to 0/1/n−1/n dimensions, which is exhaustive in dimension 3), so one cannot only work with (pseudo)scalars and (pseudo)vectors.
In any dimension, assuming a nondegenerate form, grad of a scalar function is a vector field, and div of a vector field is a scalar function, but only in dimension 3 or 7^{[2]} (and, trivially, in dimension 0 or 1) is the curl of a vector field a vector field, and only in 3 or 7 dimensions can a cross product be defined (generalizations in other dimensionalities either require vectors to yield 1 vector, or are alternative Lie algebras, which are more general antisymmetric bilinear products). The generalization of grad and div, and how curl may be generalized is elaborated at Curl: Generalizations; in brief, the curl of a vector field is a bivector field, which may be interpreted as the special orthogonal Lie algebra of infinitesimal rotations; however, this cannot be identified with a vector field because the dimensions differ – there are 3 dimensions of rotations in 3 dimensions, but 6 dimensions of rotations in 4 dimensions (and more generally dimensions of rotations in n dimensions).
There are two important alternative generalizations of vector calculus. The first, geometric algebra, uses kvector fields instead of vector fields (in 3 or fewer dimensions, every kvector field can be identified with a scalar function or vector field, but this is not true in higher dimensions). This replaces the cross product, which is specific to 3 dimensions, taking in two vector fields and giving as output a vector field, with the exterior product, which exists in all dimensions and takes in two vector fields, giving as output a bivector (2vector) field. This product yields Clifford algebras as the algebraic structure on vector spaces (with an orientation and nondegenerate form). Geometric algebra is mostly used in generalizations of physics and other applied fields to higher dimensions.
The second generalization uses differential forms (kcovector fields) instead of vector fields or kvector fields, and is widely used in mathematics, particularly in differential geometry, geometric topology, and harmonic analysis, in particular yielding Hodge theory on oriented pseudoRiemannian manifolds. From this point of view, grad, curl, and div correspond to the exterior derivative of 0forms, 1forms, and 2forms, respectively, and the key theorems of vector calculus are all special cases of the general form of Stokes' theorem.
From the point of view of both of these generalizations, vector calculus implicitly identifies mathematically distinct objects, which makes the presentation simpler but the underlying mathematical structure and generalizations less clear. From the point of view of geometric algebra, vector calculus implicitly identifies kvector fields with vector fields or scalar functions: 0vectors and 3vectors with scalars, 1vectors and 2vectors with vectors. From the point of view of differential forms, vector calculus implicitly identifies kforms with scalar fields or vector fields: 0forms and 3forms with scalar fields, 1forms and 2forms with vector fields. Thus for example the curl naturally takes as input a vector field or 1form, but naturally has as output a 2vector field or 2form (hence pseudovector field), which is then interpreted as a vector field, rather than directly taking a vector field to a vector field; this is reflected in the curl of a vector field in higher dimensions not having as output a vector field.