-
covariant (called
covectors or 1-forms)
actually pull back
under smooth functions,
meaning that the
operation ****igning the
space of
covectors to a
smooth manifold...
-
represented as n × 1
matrices (column vectors),
while covectors are
represented as 1 × n
matrices (row
covectors). When
using the
column vector convention: "Upper...
- 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...
-
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...
- ^{\sharp }.} In this extension, in
which ♭ maps p-vectors to p-
covectors and ♯ maps p-
covectors to p-vectors, all the
indices of a
totally antisymmetric tensor...
- {\displaystyle \mathbb {R} ^{n}} are
represented by
column vectors, and that
covectors (linear maps R n → R {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} }...
- ) ∗ {\displaystyle (\mathbb {R} ^{n})^{*}}
denotes the dual
space of
covectors,
linear functions v ∗ : R n → R {\displaystyle v^{*}:\mathbb {R} ^{n}\to...
-
texts the
roles are
reversed and
vectors are
defined as
linear maps from
covectors to
scalars For instance, f ( 1 + 1 ) = a + 2 r ≠ 2 a + 2 r = f ( 1 ) +...
-
covariant vector. For a pair α and β of
covector fields,
define the
inverse metric applied to
these two
covectors by The
resulting definition, although...
-
elements of the
cotangent space are
called cotangent vectors or
tangent covectors. All
cotangent spaces at
points on a
connected manifold have the same...
