Contravariant Definition and 92 Threads

Hi, I am very confused about the mathematics related to special relativity. I have understood, that a four-vector with an upper index has the form: $$A^\mu = (A^0 , A^1, A^2, A^3)$$ where lowering the index would make the indices other than the ##0##th negative: $$A_\mu = (A_0, -A^1, -A^2...

1. The laplacian is defined such that $$ \vec{\nabla} \cdot \vec{\nabla} V = \nabla_i \nabla^i V = \frac{1}{\sqrt{Z}} \frac{\partial}{\partial Z^{i}} \left(\sqrt{Z} Z^{ij} \frac{\partial V}{\partial Z^{j}}\right)$$ (##Z## is the determinant of the metric tensor, ##Z_i## is a generalized...

The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.My attemptWhat I have tried is to express this tensor...

Hello, I have a question regarding the contravarient transformation of vectors. So the formula: V'n = dx'n / dxm Vm So in words, the nth basis vector in the ' frame of reference over the mth (where m is the summation term) basis vector in the original frame of reference times the mth...

I've heard that the wavefunction as a function of x has units of square root of inverse distance, but I haven't heard an intuitive description of why this is aside from that the math works out when you integrate to get the probability. But aside from the math working out, I'm hoping to get a...

Hey, so I've been studying some math on my own and I'm really confused by this one bit. I understand what contravariant components of a vector are, but I don't understand the ways in which they transform under a change of coordinate system. For instance, let's say we have two coordinate...

We have a basis {##\mathbf{e}_1##, ##\mathbf{e}_2##, ##\dots##} and the corresponding dual basis {##\mathbf{e}^1##, ##\mathbf{e}^2##, ##\dots##}. I learned that a vector ##\vec{V}## can be expressed in either basis, and the components in each basis are called the contravariant and covariant...

I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things. Based on my understanding, the contravariant component of a vector transforms as, ##A'^\mu = [L]^\mu~ _\nu A^\nu## the covariant component of a vector...

Is there a purpose of using covariant or contravariant tensors other than convenience or ease in a particular coordinate system? Is it possible to just use one and stick to one? Also what is the meaning of mixed components used in physics , is there a physical significance in choosing one over...

The first part I'm fairly sure is just the regular gradient in polar coordinates typically encountered: $$\nabla u= \hat {\mathbf e_r} \frac {\partial u} {\partial r} + \hat {\mathbf e_\theta} \frac 1 r \frac {\partial u} {\partial \theta}$$ or in terms of scale factors: $$=\sum \hat...

In a spherical polar coordinate system if the components of a vector given be (r,θ,φ)=1,2,3 respectively. Then the component of the vector along the x-direction of a cartesian coordinate system is $$rsinθcosφ$$. But from the transformation of contravariant vector...

If we have two sets of coordinates such that x1,x2...xn And y1,y2,...ym And if any yi=f(x1...,xn)(mutually dependent). Then dyi=(∂yi/∂xj)dxj Again dyi/dxk=(∂2yi/∂xk∂xj)dxj+∂yi/∂xk Is it the contravariant derivative of a vector?? Or in general dAi/dxk≠∂Ai/∂xk

Depending on the source, I'll often see EFE written as either covariantly: $$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8 \pi GT_{\mu\nu}$$ or contravariantly $$R^{\alpha\beta} - \frac{1}{2}Rg^{\alpha\beta} = 8 \pi GT^{\alpha\beta}$$ Physically, historically, and/or pragmatically, is there a...

Homework Statement I am studying co- and contra- variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu-4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of v either by dropping perpendiculars...

I am working through a derivation of the Dirac matrix transformation properties. I have a tensor for the Lorentz transformation that is covariant on the first index and contravariant on the second index. For the derivation, I need vice versa, i.e. covariant on the second index and contravariant...

I'm going through Introduction_to_Tensor_Calculus by Wiskundige_Ingenieurstechnieken. I want to find the covariant and contravariant components of the cylindrical cooridnates. Ingerniurstechnieken sets tangent vectors as basis (E1, E2, E3). He also sets the normal vectors as another basis (E1...

Please help. I do understand the representation of a vector as: vi∂xi I also understand the representation of a vector as: vidxi So far, so good. I do understand that when the basis transforms covariantly, the coordinates transform contravariantly, and v.v., etc. Then, I study this thing...

Homework Statement I am self studying relativity. In Wikipedia under the four-gradient section, the contravariant four-vector looks wrong from an Einstein summation notation point of view. https://en.wikipedia.org/wiki/Four-vector Homework Equations It states: E0∂0-E1∂1-E2∂2-E3∂3 = Eα∂α...

This is (should be) a simple question, but I'm lost on a negative sign. So you have ##D_m V_n = \partial_m V_n - \Gamma_{mn}^t V_t## with D_m the covariant derivative. When trying to deduce the rule for a contravariant vector, however, apparently you end up with a plus sign on the gamma, and I'm...

1) I read different texts on Contravariant , Covariant vectors. 2) Contravariant - they say is like vector . Covariant is like gradient From what I see they have those vector spaces because it eventually helps get scalar out of it if we multiply contravariant by covariant Also Contravariant...

I am trying to make sure that I have a proper understanding of contravariant transformations between coordinate systems. The contravariant transformation formula is: Vj = (∂yj/∂xi) * Vi where Vj is in the y- frame of reference and Vi is in the x-frame of reference. Einstein summation...

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...

Homework Statement This is really 3 questions in one but I figure it can be grouped together: 1. The vector A = i xy + j (2y-z2) + k xz. is in rectangular coordinates (bold i,j,k denote unit vectors). Transform the vector to spherical coordinates in the unit vector basis. 2. Transform the...

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.

Homework Statement Hi I am reviewing the following document on tensor: https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Homework Equations In the middle of page 27, the author says: Now, using the covariant representation, the expression $$\vec V=\vec V^*$$...

Hey all, I'm just starting into GR and learning about tensors. The idea of fully co/contravariant tensors makes sense to me, but I don't understand how a single tensor could have both covariant AND contravariant indices/components, since each component is represented by a number in each index...

The invariant of SL(2,C) is proven to be invariant under the action of the group by the following \epsilon'_{\alpha\beta} = N_{\alpha}^{\rho}N_{\beta}^{\sigma}\epsilon_{\rho\sigma}=\epsilon_{\alpha\beta}detN=\epsilon_{\alpha\beta} The existence of an invariant of this form (with two indices...

Can someone explain to me what is the difference between covariant and contravariant vectors ? Thank You

I am reading a notes about tensor when I came across this which the notes did not elaborate more on it. As a result I don't quite understand why. Here it is : " Note that we mark the covariant basis vectors with an upper index and the contravariant basis vectors with a lower index. This may...

Not sure where to post this thread. That being said, can someone explain to me simply what covariant and contravariant tensors are and how covariant and contravariant transformation works? My understanding of it from googling these two mathematical concepts is that when you change the basis of...

Homework Statement Given that ##x_\mu x^\mu = y_\mu y^\mu## under a Lorentz transform (##x^\mu \rightarrow y^\mu##, ##x_\mu \rightarrow y_\mu##), and that ##x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu##, show that ##x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu##. Homework Equations...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 3.1 on categories and need help in understanding the contravariant functor $$\text{Hom}_R(\_, X)$$ as described in Bland, Example 13 in Ch. 3: Categories (page 76). Example 13 in Ch. 3 reads as...

I am trying to reconcile the definition of contravariant and covariant components of a vector between Special Relativity and General Relativity. In GR I understand the difference is defined by the way that the vector components transform under a change in coordinate systems. In SR it seems...

I'm working on the electromagnetic stress-energy tensor and I've found this in a book by Landau-Lifshitz: T^{i}_{k} = -\frac{1}{4\pi} \frac{\partial A_{\ell}}{\partial x^{i}} F^{k\ell}+\frac{1}{16\pi}\delta^{k}_{i} F_{\ell m} F^{\ell m} Becomes: T^{ik} = -\frac{1}{4\pi}...

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.

Hi, please help me .. How can I derivative covariant and contravariant fields? as in the attached picture Thanks.. http://www.gulfup.com/?tNXcaN w.r.t alpha

When we write contravariant and covariant indices, for example for the Lorentz transformation, does it matter if we write \Lambda^\mu\,_\nu or \Lambda^\mu_\nu ? i.e. if the \nu index is to the right of the \mu or they are at the same place with respect to left-right?

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...

Hellow everybody! A simples question: is it correct the graphic representation for covariant (x₀, y₀) and contravariant (x⁰, y⁰) coordinates of black vector?

If the notions of covariant and contravariant tensors were not introduced,what would happen?E.g. what form will the Einstein E.q. Guv=8πTuv be changed into ?

I understand energy-momentum tensor with contravariant indices, where I think I get T^{αβ}, but how do I derive the same result for T_{αβ}? Why are the contravariant vectors simply changed to covariant ones, and why does it work in Einstein's equation?

I'm confused about the difference between a contravariant and covariant vector. Some books and articles seem to say that there really is no difference, that a vector is a vector, and can be written in terms of contravariant components associated with a particular basis, or can be written in...

I think this may be a simple yes or no question. I am currently reading a book Vector and Tensor Analysis by Borisenko. In it he introduces a reciprocal basis \vec{e_{i}} (where i=1,2,3) for a basis \vec{e^{i}} (where i is an index, not an exponent) that may or may not be orthogonal...

I recently came across a very cool book called Div, Grad, and Curl are Dead by Burke. This is apparently a bit of a cult classic among mathematicians, not to be confused with Div, Grad, Curl, and All That. Burke was killed in a car accident before he could put the book in final, publishable...

Will anyone help me to under stand the covariant and contravariant vector ? And can anyone show me the derivation of dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz

Homework Statement In the oblique coordinate system K' defined in class the position vector r′ can be written as: r'=a\hat{e'}_{1}+b\hat{e'}_{2} Are a and b the covariant (perpendicular) or contravariant (parallel) components of r′? Why? Give an explanation based on vectors’ properties...

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...

I have been given the following problem: The covariant vector field is: \(v_{i}\) = \begin{matrix} x+y\\ x-y\end{matrix}What are the components for this vector field at (4,1)? \(v_{i}\) = \begin{matrix} 5\\ 3\end{matrix} Now I can use this information to solve the...