Covariant vectors Definition and 16 Threads

This is an elementary question in Analysis. I thought it might belong in "General Math", but finally settled on this rubrique. If a moderator wishes to move it to its proper rubrique, I would be grateful. I have come across three explanations for the distinction between covariant and...

Hi. The book I am using gives the following equations for the the Lorentz transformations of contravariant and covariant vectors x/μ = Λμν xν ( 1 ) xμ/ = Λμν xv ( 2 ) where the 2 Lorentz transformation matrices are the inverses of each other. I am trying to get equation 2...

Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$: $$\textbf{e}_i = \frac{\partial \textbf{r}}{\partial u^{i}} $$ And $$ \textbf{e}^i = \nabla u^{i} $$ In the primed...

Hello. I would like to check my understanding of how you transform the covariant coordinates of a vector between two bases. I worked a simple example in the attached word document. Let me know what you think.

Homework Statement I'm reading through A. Zee's "Quantum Field Theory in a nutshell" for personal learning and am a bit confused about a passage he goes through when discussing field theory for the electromagnetic field. I am well versed in non relativistic quantum mechanics but have no...

I am taking a course on GR and trying to understand Tensor calculus. I think I understand contravariant tensor (transformation of objects such as a vector from one frame to another) but I am having a hard time with covariant tensors. I looked into the Wikipedia page...

I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922 If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point)...

I know if the number of coordinates are same in both the old and new frame then A.B=A`.B` . But if the number of coordinates are not same in both old and new frame then A.B=0 means that both the vectors A and B are perpendicular. Why is it so that if the number of coordinates of both the frames...

I'm having trouble understanding those concepts in the title. Can someone explain those concepts in an easy to understand manner? Please don't refer me to a wikipedia page. I know some linear algebra and multi-variable calculus. Thank you.

Can someone explain why the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector and the derivative with respect to a covariant coordinate transforms as a contravariant 4-vector.

I remember I have read somewhere that contravariant/covariant vectors correspond to polar/axial vectors in physics, respectively. Examples for polar/axial vectors are position, velocity,... and angular momentum, torque,..., respectively. Is this right? Can I prove that, say, any axial...

Hey all, I starting to study QED along with a slew with other materials. (I read in the QED book and when I don't understand a reference I go to Jackson's E&M and work some problems out, it has been beneficial thus far!) Most of the topics are not too far fetched but I am struggling to...

Ok, so here's my problem. I just graduated with a mathematics degree and am going full force into a physics graduate program. I'm taking a course called mathematical methods for physicists, in which the first subject is tensors. Everyone else seems to be comfortable with the material, but me...

Hi all, I am new to General Relativity and I started with General Relativity Course on Youtube posted by Stanford (Leonard Susskind's lectures on GR). So first thing to understand is transformation of covariant and contravariant vectors. Before I can understand a transformation, I would...

Homework Statement I know this is an easy question, I just can't seem to grasp what I am actually doing: Let M be a manifold. Let Va be contravariant, and Wa be covariant. Show that \mu=VaWa Homework Equations (couldn't get Latex to work consistently, sorry) (1) V 'a = (dx 'a /...

If there is a contravariant vector v=aa+bb+cc with a reciprocal vector system where [abc]v=xb×c+ya×c+za×b would the vector expressed in the reciprocal vector system be a covariant vector? Is there any connection between the reciprocal vector system of a covariant vector and a...