^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] << [[Schutz - 2.8 - Basis Vectors and Basis Vector Fields|2.8 Basis Vectors and Basis Vector Fields]]| [[Schutz - 2.17 - Examples of One-Forms|2.17 Examples of One-Forms]] >> # One-Forms In this section, we go back to working on $T_P$, the [[Schutz - 2.7 - Vectors and Vector Fields#^4e794a|tangent space]] to a manifold $M$ at a point $P$. >**One Form** > A one form $\tilde{\omega}$ is a linear, real-valued functional of [[Schutz - 2.7 - Vectors and Vector Fields#^ab2ebc|vectors]], i.e. a one-form $\tilde{\omega}$ at $P$ associates with a vector $\pmb{V}$ at $P$ a real number $\tilde{\omega}(\pmb{V}) \in \mathbb{R}$. ^7e7ffc One forms are linear: ^9071be $ \tilde{\omega}(a \pmb{V} + b \pmb{W}) = a \tilde{\omega}(\pmb{V}) + b \tilde{\omega}(\pmb{W}), \quad \forall a, b \in \mathbb{R}, \pmb{V}, \pmb{W} \in T_P $ And we define their addition and multiplication by real numbers as: $ (a \tilde{\omega})(\pmb{V}) = a[\tilde{\omega}(\pmb{V})] (\tilde{\omega} + \tilde{\sigma})(\pmb{V}) = \tilde{\omega}(\pmb{V}) + \tilde{\sigma}(\pmb{V}) $ Thus, one-forms at a point $P$ satisfy the axioms of a [[Schutz - 1.5 - Linear Algebra#^c484af|vector space]], known as the dual space to $T_P$. > **Cotangent Space** > The cotangent space $T^*_P$ at point $P\in M$ where $M$ is a manifold is the dual space to the tangent space $T_P$ at $P$. It is the space of one-forms at $P$. ^0fa8ac The spaces are said to be dual to one another since one can also think of vectors as linear functions that map a one-form to a real number, $\tilde{\omega}(\pmb{V}) \equiv \pmb{V}(\tilde{\omega})$ In older treatments of tensors, vectors are often called *contravariant vectors* and one-forms are called *covariant vectors*. We will see why in [[Schutz - 2.26 - Basis Transformations|§2.26]]