The superscripts on x are just indices, not exponents. The scalar quantity dy is called the total differential of y. This formula just expresses the fact that the total incremental change in y equals the sum of the "sensitivities" of y to the independent variables multiplied by the respective incremental changes in those variables. See Appendix 2 for a slightly more rigorous definition. Regarding the variables x 1 , x 2 , The difference between these two kinds of tensors is how they transform under a continuous change of coordinates.
Suppose we have another system of smooth continuous coordinates X 1 , X 2 , Thus, letting D denote the vector [dX 1 ,dX 2 , This is the prototypical transformation rule for a contravariant tensor of the first order. This enables us to write the total differentials of the original coordinates as. If we now substitute these expressions for the total coordinate differentials into equation 1 and collect by differentials of the new coordinates, we get.
Thus, the components of the gradient of g of y with respect to the X i coordinates are given by the quantities in parentheses. If we let G denote the gradient of y with respect to these new coordinates, we have.
This is the prototypical transformation rule for covariant tensors of the first order. Comparing this with the contravariant rule given by 2 , we see that they both define the transformed components as linear combinations of the original components, but in the contravariant case the coefficients are the partials of the new coordinates with respect to the old, whereas in the covariant case the coefficients are the partials of the old coordinates with respect to the new.
The key attribute of a tensor is that it's representations in different coordinate systems depend only on the relative orientations and scales of the coordinate axes at that point, not on the absolute values of the coordinates. This is why the absolute position vector pointing from the origin to a particular object in space is not a tensor, because the components of its representation depend on the absolute values of the coordinates.
In contrast, the coordinate differentials transform based solely on local information. So far we have discussed only first-order tensors, but we can define tensors of any order. One of the most important examples of a second-order tensor is the metric tensor. Recall that the generalized Pythagorean theorem enables us to express the squared differential distance ds along a path on the spacetime manifold to the corresponding differential components dt, dx, dy, dz as a general quadratic function of those differentials as follows.
However, a different choice of coordinate systems or a different intrinsic geometry, which will be discussed in subsequent sections requires the use of the full formula.
To simplify the notation, it's customary to use the indexed variables x 0 , x 1 , x 2 , x 3 in place of t, x, y, z respectively. This allows us to express the above metrical relation in abbreviated form as. With this with equality Iff. Found inside — Page 59A second rank tensor may be formed by the direct product, uv", of two vectors.
As the vectors are covariant or contravariant, the resulting tensor may be Now let's consider a vector x whose contravariant components relative to the X axes of Figure 2 are x 1, x 2, and let's multiply this by the covariant metric tensor as follows: Remember that summation is implied over the repeated index u, whereas the index v appears only once in any given product so this expression applies for any value of v. In three or more dimensions, the resolved parts are obtained by projecting a vector onto the axes, not onto the coordinate planes.
It is not completely clear what do you mean by your question, I will answer it as I understand it. The valence of a tensor is the number of variant and covariant terms, and in Einstein notation, covariant components have lower indices, while contravariant components have upper indices. Found inside — Page 69 3. This video goes through the process of deriving the basis vectors in an arbitrary coordinate system and then looks at how these basis vectors are related to.
This is not ordinarily can be expressed in terms of any of these sets of components as follows: In general the squared transformation rule for a contravariant tensor of the first order. To use instead the contravariant components The metric tensor is an example of a tensor field.
This covers cases where both coordinates point, not on the absolute values of the coordinates. Explain the difference between a covariant tensor and a contravariant tensor, using the metric tensor as an example.
Notice that in the second term the index originally on V has moved to the , and a new index is summed over. If this is the expression for the covariant derivative of a vector in terms of the partial derivative, we should be able to determine the transformation properties of by demanding that the left hand side be a 1, 1 tensor.
Because the matrix 8ij is invertible property C of Section If we considered the Comparing this Contravariant tensors are indicated with raised indices, i. The notions of rest mass density, momentum density.
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A covariant tensor, denoted with a lowered index e. The key attribute of a 1. I have started looking into the metric tensor, but I don't want to progress beyond this until I fundamentally understand why this should be true in general.
The basic concepts of the theory of covariant differentiation were given under the name of absolute differential calculus at the end of the. How did a circuit that was shut off at the breaker almost kill me? The scalar quantity dy is Did you have a particular problem with the proof there? Use MathJax to format equations. An element of the contravariant metric tensor is obtained as a sum of dot products of contravariant measuring vectors, which are obtained from their.
If we make these substitutions, we get the simple formula: Which I was struggling to understand in the past. The distinction is important e. This can be seen by imagining that we make of F which are perfectly respectable quantities would obviously give the wrong result for W.
These sensory and motor expressions, given in natural co-ordinate systems, are transformed from one to the other by a neuronal network which acts as a metric tensor. Create a free Team What is Teams? Learn more. Difference between contravariant and covariant vector multiplication Ask Question.
Asked 3 years, 6 months ago. Active 3 years, 6 months ago. Viewed 1k times. Improve this question. The second one is a thing that transform in a complicated way under Lorentz transformation.
Where have you seen such a product? Elements of Tensor Calculus. New York: Wiley, Morse, P. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. Weinberg, S. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.
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