It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.

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Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$

Let $U = \mathbb{R}^3 \setminus \{(0,0,z) \}$ (ie $\mathbb{R}^3$ with the $z$-axis removed ) and consider $\beta$ on $U$ given by $\displaystyle \beta = \frac{x dy \wedge dz + y dz ...