Let $F: \mathbb{R}^2 \rightarrow \mathbb{R P}^2$ be the smooth map $F(x, y)=[x, y, 1]$, and let $X \in$ $\mathfrak{X}\left(\mathbb{R}^2\right)$ be defined by $X=x \partial / \partial y-y \partial / \partial x$. Prove that there is a vector field $Y \in \mathfrak{X}\left(\mathbb{R}^2\right)$ that is $F$-related to $X$, and compute its coordinate representation in terms of each of the charts defined in Example 1.5.

Homepage › Solution manuals › John Lee › Introduction to Smooth Manifolds › Problem 8-12

Problem 8-12

Let $F : ℝ^{2} \to {ℝℙ}^{2}$ be the smooth map $F (x, y) = [x, y, 1]$ , and let $X \in$ $𝔛 (ℝ^{2})$ be defined by $X = x∂ ∕ ∂y - y∂ ∕ ∂x$ . Prove that there is a vector field $Y \in 𝔛 (ℝ^{2})$ that is $F$ -related to $X$ , and compute its coordinate representation in terms of each of the charts defined in Example 1.5.

Problem 8-12

Add answer