Definition talk:Jacobian
I'm not quite sure how to put it: we need an open ball around $x$ to be able to differentiate. Possibilities:
1.have a theorem page saying an open set is a union of '$n$-cubes' (product of open intervals) and link there...then there's no need to know any topology, just take this as the meaning of the expression "open set"
2. Instead of open set, make a definition for 'product of open intervals' and restrict to the case (which isn't really much of a restriction)
3. Just assume the function is defined on all of $\R^n$ and ignore the problem altogether
4. Show that an interior point of a set in $\R^n$ is one contained in some ball contained in the set, then make $x$ an interior point and link to the just mentioned page. (a bit complicated, and kind eccentric)
5. Leave the topology (unnecessary) --Linus44 (talk) 19:58, 29 September 2012 (UTC)
- We've already got on the Definition:Open Rectangle page, a generalisation of the concept of Definition:Real Interval to n dimensions.
- We also have a considerable quantity of material demonstrating that the real number space is a metric space under the usual topology and others (but the Euclidean topology is the one relevant here, one supposes). Then of course we have the links to demonstrate that a metric space is a topology. Thus we have the underpinnings to prove that a multi-dimensional real interval is an open set in the topological sense if we needed to demonstrate the topological structure underneath - but I contend that we don't need this in this context, as it gets in the way of the fact that this definition is on a Euclidean space of real numbers. Simply linking to Open Set (Topology) is far too loose an option. Even referring to an "open ball" in this context is overdoing it, imo. --prime mover (talk) 20:19, 29 September 2012 (UTC)
- Maybe it is an idea to put this as Jacobian matrix, differentiating it from the Jacobian determinant important in integral calculus (esp. in $\R^n$ with all that fancy exterior algebra stuff and top forms). --Lord_Farin (talk) 21:17, 29 September 2012 (UTC)
- Done. This is the standard technique for such multi-concept definitions. --prime mover (talk) 05:31, 30 September 2012 (UTC)