a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form...

exact differential forms Complex differential form Vector-valued differential form Equivariant differential form Calculus on Manifolds Multilinear form Polynomial...

connection and vector-valued differential forms. A 'naïve' attempt to define the derivative of a tensor field with respect to a vector field would be...

derivative. A connection form associates to each basis of a vector bundle a matrix of differential forms. The connection form is not tensorial because...

any differential k-form ω and any vector-valued form s. This may also be viewed as a direct inductive definition. For instance, for any vector-valued differential...

In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the...

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional...

and d ν {\displaystyle d\nu } the differential of ν {\displaystyle \nu } regarded as a vector-valued differential form, and the brackets denote the metric...

multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used...

Frölicher–Nijenhuis bracket is an extension of the Lie bracket of vector fields to vector-valued differential forms on a differentiable manifold. It is useful in the...

Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including units of measurement...

mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an...

linear algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation...

determines the curve. A parametric Cr-curve or a Cr-parametrization is a vector-valued function γ : I → R n {\displaystyle \gamma :I\to \mathbb {R} ^{n}} that...

is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization...

vector of a function F at a point is explicitly as the limiting value of a vector-valued surface integral around a shell enclosing p divided by the volume...

{\displaystyle df} amalgamates these forms into a single object and is therefore an instance of a vector-valued differential form. The chain rule has a particularly...

with a differential form on the target manifold. Covariant derivatives or differentials provide a general notion for differentiating of vector fields...

In vector calculus, the gradient of a scalar-valued differentiable function f {\displaystyle f} of several variables is the vector field (or vector-valued...

In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information...

differential geometry, where it is used to define differential forms. Differential forms are mathematical objects that evaluate the length of vectors...

length) and direction. Euclidean vectors can be added and scaled to form a vector space. A vector quantity is a vector-valued physical quantity, including...

elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field...

unit normal vector field n to f(V), one defines the following objects as real-valued or matrix-valued functions on V. The first fundamental form depends only...

In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold...

covariantly (like basis vectors) and the ones that transform contravariantly (like components of a vector and differential forms) are "almost the same"...

first-order homogeneous linear partial differential equations. In modern geometric terms, given a family of vector fields, the theorem gives necessary and...

of a differential equation is a function that satisfies the equation. The solutions of a homogeneous linear differential equation form a vector space...

The values of j may be complex numbers, or in fact complex square matrices, corresponding to the possibility of vector-valued automorphic forms. The...

(orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology (intersection forms of manifolds, especially...