- 14
- 0
In how many dimensions does a quark manifest?
What is the difference between a contravariant tensor and a covariant tensor?
What is the difference between a contravariant tensor and a covariant tensor?
Not "components", indices: the component T^{0}_{1} of some tensor T^{a}_{b} has both contravariant and covariant indices.Originally posted by HallsofIvy As far as "covariant" and "contravarient" are concerned...one should talk about "convariant" and "contravarient" COMPONENTS of a tensor.
Rather than "coefficient", we usually say component, or in the case of a 2nd rank tensor, matrix element.Originally posted by HallsofIvy g_ij represents its contravarient coefficients...
Generally speaking, it's spaces and not the coordinate systems defined on them that are viewed as being endowed with geometrical properties.Originally posted by HallsofIvy In a Euclidean coordinate system (theta= pi/2) "parallels" and "perpendiculars" are exactly the same and so these coordinates are exactly the same. If theta is not pi/2, they are not the same.
Tensoriality is defined in terms of transformation properties under coordinate changes, not in terms of the geometry of the spaces on which they're defined, so the term "euclidean" in the way you've used it doesn't apply to tensors. However, in the special case of the metric tensor, the term "euclidean" is used to indicate it's positive-definiteness as an inner product. But even with tensors on spaces with no geometry, contravariant and covariant indices can be differentiated in the sense of being dual to each other, like dual vector spaces.Originally posted by HallsofIvy In Euclidean tensors, there is no difference between covariant and contravariant components.