^^ [[Schutz - 2 - Differentiable Manifolds and Tensors|2. Differentiable Manifolds and Tensors]] << [[Schutz - 2.27 - Tensor Operations on Components|2.27 Tensor Operations on Components]] | # Functions and Scalars A *scalar* is defined as a $(0,0)$ tensor, i.e. a function on the manifold whose definition does not depend upon the choice of any particular basis. For example, the contraction $V^i\omega_i$ is a scalar, since its value is independent of the particular basis in which the components are computed. On the other hand, each component of a vector $V^i$ is also a function on the manifold, having a numerical value at each point. It is *not* a scalar because its value depends on the basis. Whether a thing is a 'scalar' or simply a 'function' depends on its interpretation when the basis is changed, rather than on its actual value.