There are several ways to create offset section views in a SOLIDWORKS drawing. We will use an offset section to circumvent the need for multiple section views. Because the center of the hole and the slotted pocket are not coplanar, we would have to use multiple standard sections to detail both of these features. We want to use section views to show the details of the machined features of the part. We will start with a top view and an isometric view of the part, a mounting fixture. This article will discuss the different ways we can create these powerful drawing views in SOLIDWORKS. What may sometimes require two or even three linear section views can sometimes be accomplished in a single offset section view. You may also be interested in looking at Hartshorne chapter II exercise 5.18.Offset section views in SOLIDWORKS Drawings allow the user to create efficient, informative sections of a model. This is called the espace étalé and is discussed here and somewhere in Hartshorne chapter II section 1. Conversely, if we have a sheaf on a space $X$, it is possible to construct a space $Y$ and a map $Y \to X$ such that the "sections" of the sheaf correspond to the actual sections of the map $Y \to X$.
The corresponding sheaf is the one which maps open sets $U$ to the set of sections of the bundle over $U$. Whenever we have a bundle, we can form a sheaf out of it. Similarly a global section of a bundle is the same as a bunch of local sections (which, again, are just functions) that "agree" on the overlaps, where now "agree" does not mean literally agree, but "agree after a twist", where the "twists" comes from the transition functions of the vector bundle. A global function on a manifold is the same as a bunch of local functions that literally agree on the overlaps. A section of a trivial bundle is just a function $U \to F$. A trivial bundle with fiber $F$ looks like the projection map $U \times F \to U$. To your third question, I think the observation that $\Gamma(-,Y)$ forms a sheaf on $X$ gives a nice context in which to think of sections $X$ to $Y$: they "live in" the sheaf $\Gamma(-,Y)$ as its globally defined elements.īundles are usually defined as being locally trival thingamajigs. More unfortunate is the annoying coincidence that when dealing with schemes the projection map from the espace étalé happens to be an étale morphism, because it is locally on its domain an isomorphism of schemes, a much stronger condition.$\Big)$ This is unfortunate, because the espace étalé has very little to with with étale cohomology.
However, the French word "étalé" means "spread out", whereas "étale" (without the second accent) means "calm", and they were not intended to be used interchangeably in mathematics. $\Big($ Unfortunate linguistic warning: Many people incorrectly use the term "étale space". This explains the otherwise bizarre tradition of writing $\Gamma(U,F)$ instead of the the more compact notation $F(U)$. $\Gamma(-,Y)$ actually forms a sheaf of sets on $X$.Ĭonversely, given any sheaf of sets $F$ on a space $X$, one can form its espace étalé, a topological space over $X$, say $\pi: \acutet(F))$. maps $U\to Y$ such that the composition $U \to Y\to X$ is the identity (thus necessarily landing back in $U$). For $U\subseteq X$ open, the notation $\Gamma(U,Y)$ denotes sections of the map $\pi$ over $U$, i.e. The word "over" is used to activate the tradition of suppressing reference to the map $\pi$ and refering instead to the domain $Y$. Say $\pi: Y\to X$ is a space over $X$ (intentionaly vague). To your second question, I generally take the "right-inverse" or "pre-inverse" definition from category theory, because it relates back to others in the following precise way: Thus locally a section just looks like a function with codomain $T$, which is often required to be nice. $U\subset X$) isomorphic to some product $U\times T$, then we can locally identify the fibres with $T$.
#Section line geometry free
If one is talking about locally free / locally trivial bundles, meaning $E$ is locally (over open sets at each point $x\in X$, it takes value in the fibre (This is a fairly selective use of the word "function" which used to confuse me.) A section $\gamma$ of a (some-kind-of) bundle $E\to X$ is thought of as a "generalized function" on $X$ by thinking of it as a funcion with "varying codomain", i.e. To your first question, "function on a space" $X$ usually means a morphism from $X$ to one of several "ground spaces" of choice, for example the reals if you work with smooth manifolds, Spec(A) if you work with schemes over a ring, etc.
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