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- Thread starter NotASmurf
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Matterwave

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Can you explain your notation? What are "x',y',x,y" supposed to be in this picture?

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Matterwave

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That is some seriously confusing notation. What book is this?

A change of coordinates from (x,y) to (x',y') looks simply like ##x'=x'(x,y),~y'=y'(x,y)## so these are simply functions of the old coordinates. Vector and one form components transform similarly to the picture you uploaded, but I've never seen that kind of notation. Usually the notation is such that for a vector: $$V^{\alpha'} = \sum_\beta \frac{\partial x^{\alpha'}}{\partial x^\beta} V^\beta$$ And for a one form: $$\omega_{\alpha'}=\sum_\beta \frac{\partial x^\beta}{\partial x^{\alpha'}}\omega_\beta$$

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Fredrik

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Those equations and the corresponding ones for y' *define* covariant and contravariant in this (obsolete and horrible) approach to tensors, so it's a bit odd to ask why the equations look the way they do. It does however make sense to ask why those terms are used. The idea is that in the covariant case, the matrix that transforms ##\begin{pmatrix}x\\ y\end{pmatrix}## to ##\begin{pmatrix}x'\\ y'\end{pmatrix}## is the same matrix that transforms the basis vectors (hence the term *co*variant), and in the contravariant case, the matrix that does the transformation is the inverse of the matrix that transforms the basis vectors (hence the term *contra*variant).

The modern approach starts with a vector space V. Its dual space V* is defined as the vector space of linear maps from V into ℝ. Now suppose that ##(e_1,\dots,e_n)## and ##(e_1',\dots,e_n')## are ordered bases for V. There must exist numbers ##M^j_i## such that ##e_i'=M^j_i e_j## (each of the primed basis vectors can be expressed as a linear combination of the unprimed basis vectors). Now we can examine the relationship between the components of an arbitrary ##v\in V## with respect to these two ordered bases. It turns out that it's given by ##v^i{}'=(M^{-1})^i_j v^j##. Because of this, the n-tuple of components is said to "transform contravariantly".

**Edit:** I fixed a typo in the paragraph above after it was pointed out by Matterwave below.

There's a simple way to use an ordered basis for V to define an ordered basis for V*. Because of this, it makes sense to ask for the relationship between the components of an arbitrary ##f\in V^*## with respect to the two ordered basis for ##V^*## that are defined from the two ordered bases for ##V## mentioned above. It turns out that the relationship is given by ##f_i'=M_i^j f_j##. Because of this, the n-tuple of components is said to "transform covariantly".

You can find many of the details of this approach in this post. I also recommend chapter 3 in "A first course in general relativity" by Schutz. It's a nice introduction to this stuff.

The modern approach starts with a vector space V. Its dual space V* is defined as the vector space of linear maps from V into ℝ. Now suppose that ##(e_1,\dots,e_n)## and ##(e_1',\dots,e_n')## are ordered bases for V. There must exist numbers ##M^j_i## such that ##e_i'=M^j_i e_j## (each of the primed basis vectors can be expressed as a linear combination of the unprimed basis vectors). Now we can examine the relationship between the components of an arbitrary ##v\in V## with respect to these two ordered bases. It turns out that it's given by ##v^i{}'=(M^{-1})^i_j v^j##. Because of this, the n-tuple of components is said to "transform contravariantly".

There's a simple way to use an ordered basis for V to define an ordered basis for V*. Because of this, it makes sense to ask for the relationship between the components of an arbitrary ##f\in V^*## with respect to the two ordered basis for ##V^*## that are defined from the two ordered bases for ##V## mentioned above. It turns out that the relationship is given by ##f_i'=M_i^j f_j##. Because of this, the n-tuple of components is said to "transform covariantly".

You can find many of the details of this approach in this post. I also recommend chapter 3 in "A first course in general relativity" by Schutz. It's a nice introduction to this stuff.

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Thank you so much.

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Matterwave

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There must exist numbers ##M^j_i## such that ##e_i'=M^j_i e_i##

I believe you mean ##e_i'=M^j_i e_j## :)

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Confusion, in the video teaching me this they say thatin the contra-variant case, the matrix that does the transformation is the inverse of the matrix that transforms the basis vectors

equation 4 is the contravarient one and that equation 5 (the inverse) is the covariant one, your intuition made sense but this confused me now, his notation isn't helping.

Also covarient is the one that is reference frame independent right? Apologies for newbie-ness.

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Matterwave

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All tensors are "reference frame independent", in the appropriate sense.

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Fredrik

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In my notation, the first one is ##T_y^{mn}=(M^{-1})^m{}_r(M^{-1})^n{}_s T_x^{rs}## and the second one is ##T_{mn}(y)=M^r{}_m M^s{}_n T_{rs}(x)##.Confusion, in the video teaching me this they say that

equation 4 is the contravarient one and that equation 5 (the inverse) is the covariant one, your intuition made sense but this confused me now, his notation isn't helping.

Elements of ##V## and elements of ##V^*## are all independent of the frame (=the ordered basis). But their components with respect to an ordered basis are not.Also covarient is the one that is reference frame independent right?

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Fredrik

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Yes, thanks. I wrote that post very quickly. I have edited that typo now. Also, it would have been even better to write ##e_i'=M^j{}_i e_j##, because when we start using the metric to raise and lower indices, we're going to have to keep track of the horizontal positions of the indices.I believe you mean ##e_i'=M^j_i e_j## :)

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one form? what does that mean?

All tensors are "reference frame independent", in the appropriate sense.

Thanks for help so far, I liked your definition of contravarient.

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Fredrik

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There's a vector space ##T_pM## associated with each point ##p## in a smooth manifold ##M##. An element of ##T_pM## is called a tangent vector at ##p##. An element of ##T_pM^*## is called a cotangent vector at ##p##. A vector field on ##M## is a function that associates a tangent vector at ##p## with each ##p\in M##. A cotangent vector field on ##M## is a function that associates a cotangent vector at ##p## with each ##p\in M##. A cotangent vector field is also called a 1-form (for reasons that I will not go into here).one form? what does that mean?

Thanks for help so far, I liked your definition of contravarient.

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Matterwave

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one form? what does that mean?

Thanks for help so far, I liked your definition of contravarient.

For now, you can just think of it as simply another way to say "co variant vector". The language of forms (there are 0-forms, 1-forms, 2-forms, etc.), mostly worked through by Elie Cartan, is a very powerful one in differential geometry. It gives a rigorous definition of integration on manifolds, a connection and congruence independent definition of a derivative (called an exterior derivative), as well as a generalization of the fundamental theorem of calculus (called Stoke's theorem) on manifolds. But until you get to that, "one form" will just be another word for "co variant vector".

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