Elements of tensor calculus
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Scientific Research An Academic Publisher. Lichnerowicz, A. Methuen, London. We show why the two equations must have exactly the same sets of solutions, whereby this classical statistical theory CST and nonrelativistic quantum mechanics may differ only in their interpretations of the same quantitative results. We identify some of the different interpretations. We show that the results also imply nonrelativistic Lagrangian classical mechanics and the associated Newtonian laws of motion.
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We prove that the CST applied to a nonrelativistic rigid rotator yields spin angular momentum operators that obey the quantum commutation rules and allow both integer and half-odd-integer spin. We also note that the CST applied to systems of identical massive particles is mathematically equivalent to nonrelativistic quantum field theory for those particles. Related Articles:. System Simulation and Analysis. Plant Modeling for Control Design. High Performance Computing. Robert J. The Tensor subpackage of the DifferentialGeometry package supplants the now-deprecated tensor package in Maple.
The tensor package made essential use of the also-deprecated linalg package, so although worksheets that used these deprecated packages still work, it is imperative to move to using the new formalisms in the Tensor package. This article is a "survivor's guide" for implementing tensor calculus in the new Tensor package. It explains the constructs in this package from the perspective of classical i.
The DifferentialGeometry package itself contains some 34 command but also six subpackages, themselves contributing to a total of some commands. These subpackages, and a measure of their "size" are listed in Table 1. The complexity of mastering the DifferentialGeometry package is further increased when subpackages such as GroupActions themselves have subpackages MovingFrames.
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A comprehensive tutorial in the complete DifferentialGeometry package would require more than a textbook, so clearly, this is not our ambition here. Fortunately, the package itself contains two sets of useful tutorials, a comprehensive collection of Lessons , and a set of Tutorials. Each lesson also contains a set of exercises that range in difficulty from simple computations to programming problems.
Solutions are given. The tutorials present specialized applications of the DifferentialGeometry package. Our more modest goal for this article is to show how to enter covariant and contravariant tensors, compute their covariant derivatives, obtain the equations of parallel transport and geodesics, and compute the basic tensors of general relativity. Declaring the Frame in DifferentialGeometry. Before a vector or tensor can be entered, a frame must be declared by stating its variables and giving it a name. For example, to declare as the Cartesian space with variables and , execute. The default behavior for DifferentialGeometry is that from this point onward, the prompt would be modified to display the frame name for as long as that frame were the active one.
Thus, without our having suspended this default with the Preferences command in the Initializations section, the prompt following the DGsetup command would be the one shown in Table 2. We have elected to suspend this default behavior for three reasons. First, interactive editing of a worksheet can create a confusing display. If in a section where one frame name appears in all prompts, a new frame is defined, the subsequent prompts that show the earlier name do not change to show the new name. Only a "new" prompt will display the new frame name.
The result is a worksheet with prompts showing different frame names where they might not be relevant. Such misplaced prompts have to be deleted manually if they are not to provide incorrect information. Second, these modified prompts are persistent - they cannot be removed by any Maple command. They have to be removed by deletion. The Maple interface command that modifies prompts does not cascade the change through existing prompts.
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It only modifies new prompts. Finally, if the commands are executed in Document Blocks, and not at prompts, there will be no visible prompt to modify. Thus, the modified prompt is a worksheet paradigm that does not carry through all of Maple usage.
For all these reasons, we will not have frame names visible in our prompts. Anyone who had taken even a perfunctory dip into the waters of tensor calculus knows there are two words, covariant and contravariant , that must be faced. We will not be able to enter a tensor in the Tensor package without making the distinction between these two terms. Using the Einstein summation convention repeated indices, one raised and one lowered, are summed , Table 3 defines contravariant and covariant vectors.
Vector Type.
In the rightmost column of Table 3 uses the notation and for components expressed in the -coordinate system, but and for components in the -coordinate system. Texts also denote the new coordinate system bu the use of an overbar on the component, or a prime on the left side of the variable. The components and are the contravariant and covariant components, respectively, of the vector V. The basis vectors and are reciprocal, so that. Thus, an orthonormal basis is self-reciprocal. That is why the distinction between contravariant and covariant basis does not matter in Cartesian spaces.
Elements of Tensor Calculus
If is the mapping from to via functions of the form , then the gradient vectors are the rows of the Jacobian matrix , where the upper index is interpreted as a row index, and the lower index , as a column index. If is the mapping from to via functions of the form , then the tangent vectors are the columns of the Jacobian matrix. To facilitate the implementation of the contravariant transformation law, writing the components as a column vector v means the sums with the Jacobian matrix are along a row and across the columns of the matrix.
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Hence, the matrix product implements the contravariant transformation. Writing the components as the row vector w means the sums with the Jacobian matrix are down a column but across the rows. Hence, the matrix product implements the covariant transformation.
This inherent distinction between tangent bases and normal bases induces the distinction between contravariant and covariant. Using column and row vectors to express this difference is a convenient visual device in classical tensor calculus. If is a vector space, say, with basis , then a rank-two tensor is a multilinear object from , the direct product of with itself, having doubly-indexed basis objects.
The tensor is actually the object.
See also weyl. In what amounts to an abuse of this notation, itensor also allows kdelta to have 2 covariant and no contravariant, or 2 contravariant and no covariant indices, in effect providing a co ntra variant "unit matrix" capability. It is computed only the first time it is called. Marsden and T. Before a vector or tensor can be entered, a frame must be declared by stating its variables and giving it a name.
Of course, the are the contravariant components of the tensor; and just as for vectors, there would be the equivalent covariant components,. There are even mixed tensors that transform contravariantly in one index but covariantly in another. In actual practice, one manipulates just the components of the tensor, and almost never explicitly exhibits the basis objects.
However, in the DifferentialGeometry package, vectors and tensors require an explicit use of the basis objects. The basis for could be entered as. Alternatively, enter such expressions in text linear, 1D mode and convert to math mode via the Context Menu. The reciprocal or dual basis is then. In actual fact, is considered a differential form , more in keeping with the modern approach to differential geometry.
Representing Vectors as DifferentialGeometry Objects. In the DifferentialGeometry package, the contravariant vector whose components are is given by. The evalDG and the DGzip commands are two of the simpler ways to create an object whose data structure is intrinsic to the DifferentialGeometry package. When using the evalDG command, the asterisk is the explicit multiplication operator.