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Statements
Definition. (@Bac12, Ch. 3) Let be a point in and consider the Euclidean tangent space , the linear function
\omega:&T_p\mathbb{R}^n&\to&\mathbb{R}\\ &v&\mapsto&v_1\;dx_1+v_2\;dx_2+\cdots+v_n\;dx_n \end{array}$$ is said to be a *1-form*. ## Interpretation A 1-form maps the coordinates $dx$ and $dy$ of the tangent passing through the point $p\in T_p\mathbb{R}^2$ into the vector $(a,b)$. ## Example 1. Let $\omega(\langle dx, dy\rangle)=-dx+4dy$. a. Compute $\omega(\langle 1, 0\rangle)$, $\omega(\langle 0, 1\rangle)$ and $\omega(\langle 2, 3 \rangle)$. b. What line does $\omega$ project vectors onto? a. $\omega(\langle 1, 0\rangle)=-1$, $\omega(\langle 1, 0\rangle)=4$ and $\omega(\langle 2, 3 \rangle)=10$. b. Onto $\langle -1, 4\rangle$.