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marcus

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I have been unable to read the differential forms thread but wanted to, and now I have discovered how to make it legible on my browser. Lethe has been using Times Roman font and for some reason most of the symbols come thru as boxes for me in that font. So as an experiment I have quoted a Lethe post and removed the Font specification, setting that back to default. Most of the symbols now come thru for me altho a couple I see here (& sdot , & nabla ) still do not

I'm wondering if anyone else was discouraged earlier by seeing all those boxes and no being able to tell what symbols they stood for.

I'm wondering if anyone else was discouraged earlier by seeing all those boxes and no being able to tell what symbols they stood for.

Originally posted by lethe in diff forms thread

at this point, i will stop using classical vector notation. i will write the directional derivative asv^{μ}∂_{μ}ƒ, where ∂_{μ}is shorthand for ∂/∂x^{μ}, andx^{μ}is one of the coordinates, and μ is a number that ranges over the number of dimensions of the manifold, from 0 to n-1 usually. so there will be n different coordinates for an n dimensional manifold. andv^{μ}is going to be associated with the μ-th component of the vector, to be defined. and even though i didn t write it, i meant for that to be a summation:v⋅∇ƒ = ∑v^{μ}∂_{μ}ƒ =v^{μ}∂_{μ}ƒ. i just leave off the ∑ from now on. every time you see an equation with the same letter as a superscript and a subscript, you should sum over that index.

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we define the vector to be that operator. this is how it operates on a function:

v(ƒ) =v^{μ}∂_{μ}ƒ (2)

since this is independent of the function that i want to operator on, let me just write the vector operator:

v=v^{μ}∂_{μ} (3)

and this is the point of this post. a tangent vector is defined to be/associated with/thought of as a differential operator.vis the vector, andv^{μ}are the coordinate components of the vector, and ∂_{μ}are the coordinate basis vectors of the tangent space. the vector itself is coordinate independent, but the components are not, ....

OK, it should be easy to show that the set of tangent vectors, thusly defined, satisfy the axioms of the vector space. i will call this vector space TM_{p}. that is, the tangent space to the manifold M at the point p is TM_{p}. for an n dimensional manifold, the tangent space is always an n dimensional vector space.

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