Vector calculus

geometric object that has magnitude (or length) and direction

foundational law of electromagnetism

thumb|300px|The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).

calculus of vector-valued functions

assignment of a vector to each point in a subset of Euclidean space

500px|thumb|upright=1.75|alt= A vector field with diverging vectors, a vector field with converging vectors, and a vector field with parallel vectors that neither diverge nor converge|The divergence of different vector fields. The divergence of vectors from point (x,y) equals the sum of the partial derivative-with-respect-to-x of the x-component and the partial derivative-with-respect-to-y of the y-component at that point: \nabla\!\cdot(\mathbf{V}(x,y)) = \frac{\partial\, {V_x(x,y){\partial{x+\frac{\partial\, {V_y(x,y){\partial{y

in geometry, an object that is perpendicular to a given object, vector perpendicular to a curve or surface

Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus, flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over a surface.

differential operator describing the rotation at a point in a 3D vector field

In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is also a fluid. The properties that are carried with the advected substance are conserved properties such as energy. An example of advection is the transport of pollutants or silt in a river by bulk water flow downstream. Another commonly advected quantity is energy or enthalpy. Here the fluid may be any material that contains thermal energy, such as wat

right|100px|thumb|Del operator,represented bythe nabla symbol Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by ∇ (the nabla symbol). When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vec

definite integral of a scalar or vector field along a path

theorem in vector calculus

visual aid to depiction of a vector field

concept in vector analysis and physics

a vector field whose curl is a given vector field

function valued in a vector space; typically a real or complex one

thumb|right|A loop of wire (black), carrying a electric current|current I, creates a [[magnetic field B (blue). If the position and current of the wire are reflected across the plane indicated by the dashed line, the magnetic field it generates would not be reflected: Instead, it would be reflected and reversed. The position and current at any point in the wire are "true" vectors, but the magnetic field B is a pseudovector.]]

In vector calculus a solenoidal vector field is a vector field v with divergence zero at all points in the field:

addition of vectors

mathematical series approximating an angle-dependent function

theorem

concept in vector calculus

mathematical identities

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restatement of Newton's law of universal gravitation

thumb|170px|Parallel plane segments with the same orientation and area corresponding to the same bivector .

rate of transfer of energy through a surface

vector field representation in 3D curvilinear coordinate systems

in meteorology, the rate of change of shape of fluid bodies

amount of power radiated through a given area

method for visualizing vector fields

differential operator used in vector calculus