Plato’s cave and differential forms
نویسندگان
چکیده
منابع مشابه
Differential Forms and Integration
a f(x) dx (which one would use for instance to compute the work required to move a particle from a to b). For simplicity we shall restrict attention here to functions f : R → R which are continuous on the entire real line (and similarly, when we come to differential forms, we shall only discuss forms which are continuous on the entire domain). We shall also informally use terminology such as “i...
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Definition 1. An m-linear function f which maps the m-fold cartesian product V m of a vector space V into some other vector space W is called alternating if f(v1, . . . , vm) = 0 whenever v1, . . . , vm ∈ V and vi = vj for i 6= j. We let ∧ (V ,W ) be the vector space of mlinear alternating functions mapping V m intoW . We then define ∧m V by the property that if f ∈ ∧m(V,W ) then there exists a...
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Remark. It is also possible to define tensors over a complex vector space (with no essential modifications in the definitions and results below, except that the special Euclidean case no longer applies). This is important in many other applications (including complex manifolds), but will not be considered here. 1.1. The dual space. Let V ∗ = hom(V ;R) be the dual of V , i.e. the vector space of...
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Differentials are an important topic in algebraic geometry, allowing the use of some classical geometric arguments in the context of varieties over any field. We will use them to define the genus of a curve, and to analyze the ramification of morphisms between curves. Although differentials remain important for arbitrary varieties, we will restrict our treatment to the case of nonsingular varie...
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We’ve hit the big three theorems of vector calculus: Green’s, Stokes’, and the Divergence Theorem (although Green’s theorem is really a pretty obvious special case of Stokes’ theorem). But it’s an amazing fact that all of these theorems are really special cases of an even larger theorem that unifies all that we’ve studied so far. This new theory also extends what we’ve done to spaces of arbitra...
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ژورنال
عنوان ژورنال: Geometry & Topology
سال: 2019
ISSN: 1364-0380,1465-3060
DOI: 10.2140/gt.2019.23.3141