As those know who learned precalculus from Euler, (Intro. to Analysis of the Infinite, paragraphs ##174-175##, one can deduce from his complex series for the trig functions, that in general,
$$1/m^3 - 1/(2n-m)^3 + 1/(2n+m)^3 - 1/(4n-m)^3 + 1/(4n+m)^3 - ....
= (k^2+1)π^3/(8n^3k^3),$$ where ##k =...
the thing that puzzled me is that the original statement was true in all my examples (since the hypothesis implies the left part of the curve lies above its tangent line,) but the rearranged one was not. Then I finally noticed you had divided your inequality by the negative number (x-a), which...
Everything here looks correct to me, (except for the unsigned generic statements appended at the end), but I suggest looking at it a little more intrinsically.
I.e. given V,W there is a bilinear pairing L(V,W)xV-->W taking <f,v> to f(v). Fixing f, the set of all v such that f(v) = 0 is the...
Orodruin to the rescue!
As Orodruin makes clear, a map from T to T*, (where T is the tangent space), corresponds to a tensor of type (0,2), say by rule 1), post #31, since L(T,T*) ≈ T*(tens)T*. Thus its inverse is a map from T* to T, and by the isomorphism L(T*,T) ≈ T**(tens)T ≈ T(tens)T, the...
To someone like me, i.e. me, who knows nothing of this, and never heard of Einstein's field equations before, wikipedia looks actually useful.
It is an equation between two tensors of type (0,2). (I have been calling them type (2,0)), i.e. 2-covariant tensors, linear combinations of...
Ok, so I think of it this way now: in physics there are various operations on (fields of) vectors taking them linearly or multi-linearly to other (fields of) vectors, and it is useful for calculations to represent these operations as tensor fields, hence entirely in terms of tensor products of...
Ah yes, proceeding from Orodriuin's guidance, we can see what kind of tensor the Riemann curvature should be:
We need the basic insight that the map VxV-->V(tens)V sending <v,w> to v(tens)w, is bilinear, and that thus composing with any linear map out of V(tens)V yields another bilinear map...
You are reminding me of how stunned I was when my professor asked why I was so sure, when given a function f(x), that x was the variable and f the function. Why not define x(f) to be f(x), and then f was the variable?
I.e. if F is the space of all k valued functions on the set X, then we have...
well I was enjoying the excerpt of Biennow on amazon https://www.amazon.com/Mathematical-Methods-Physics-Engineering-Mattias-ebook/dp/B086H3LMZF/?tag=pfamazon01-20
until it ran out just starting tensors, but discovered that physicists like tensors so much they use them to represent objects that...
Well a quick look at Wikipedia shows that the output of a tensor can be more general than a function, e.g. the Riemann curvature tensor ( a slight elaboration on the formula in post #21) takes a triple of vector fields and outputs a vector field.
Namely the commutator of two vector fields is...
I am a mathematician, and to me, if you already know what the tangent bundle is, and its dual the cotangent bundle, and hence know what tangent and cotangent vector fields are, then the 20 page explanation in Spivak's chapter 4, of his Comprehensive introduction to differential geometry, is as...
quantifiers are actually essential for precise statements. recall existential quantifiers affirm a statement for at least one element of a given set, while universal quantifiers do so for all such elements.
although it may sound otherwise, existential statements are often stronger than...
Excellent, except to be picky, you have omitted some existential quantifiers, e.g. in line minus 17, it should say "there exists h such that". Of course also in line -16 there is an implied universal quantifier. nice work overall.
I agree especially with points made by Choppy and CrysPhys: first of all, the purpose of writing the thesis, for most of us, is to strengthen the author's research chops to the point of being ready to do, and write up, competitive research - i.e. it is a learning experience for the author of...