Definition:Graph Parametrization

Definition

Let $U \subseteq \R^n$ be an open subset of $n$-dimensional Euclidean space.

Let $f : U \to \R^k$ be a real function.

Let $\map \Gamma f$ be the graph of $f$.

Let $\gamma_f : U \to \R^n \times \R^k$ be a mapping such that:

$\map {\gamma_f} u = \tuple {u, \map f u}$

Then $\gamma_f$ is called the graph parametrization (of $\map \Gamma f$).

Sources

• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces. Topological Embeddings
• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): $\S 1.1$: Smooth Manifolds. Topological Manifolds
• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics