- mathematics, a
linear form (also
known as a
linear functional, a one-form, or a
covector) is a
linear map from a
vector space to its
field of
scalars (often, the...
-
vectors are
often just
called vectors and
covariant vectors are
called covectors or dual vectors. The
terms covariant and
contravariant were introduced...
-
where v is the
vector and v i are its
components (not the ith
covector v), w is the
covector and wi are its components. The
basis vector elements e i {\displaystyle...
- In
differential geometry, a one-form (or
covector field) on a
differentiable manifold is a
differential form of
degree one, that is, a
smooth section of...
-
applied to
vector fields and
covector fields by
applying them to each point. Hence, if X is a
vector field and ω is a
covector field, X ♭ = g ( X , ⋅ ) {\displaystyle...
- (with n-dimensional indices,
summation implied). A
covariant vector, or
covector, is a
system of
functions v k {\displaystyle v_{k}} that
transforms under...
-
length of (or
angle between)
covector fields; that is,
fields of
linear functionals. To see this,
suppose that α is a
covector field. To wit, for each point...
- of a
covector field along a
vector field v is
again a
covector field. Once the
covariant derivative is
defined for
fields of
vectors and
covectors it can...
- ) ∗ {\displaystyle (\mathbb {R} ^{n})^{*}}
denotes the dual
space of
covectors,
linear functions v ∗ : R n → R {\displaystyle v^{*}:\mathbb {R} ^{n}\to...
-
obtain a
covector: g i j v j = v i {\displaystyle g_{ij}v^{j}=v_{i}} and this is what we mean by
lowering the index. Conversely,
contracting a
covector with...
