Computer algebra systems for working with tensors
Write this article made me unable to leave comments without having an invite. I wanted to write to the author of the article, but did not find the e-mail. Here, in fact, the article itself, even a whole series of articles devoted to tensor calculus.
In comments to this article, ZloAlien asked about mathematical packages for working with tensors, to which the author maisvendoo told him that Maple 18 might be, and attached a picture with the output of tensor package functions. Here, actually, by this I could not pass by.
Computer algebra system (CAS ) computer algebra systems (CAS ) are used for symbolic computations. They are not few, but we will focus on those with whom I had to face, just when looking for the most suitable for myself.
So, in order:
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Developed in the late 80s by Stephen Wolfram contains, among other things, the Tensors in Physics package for working with tensors. Judging by the description, it can perform basic actions on tensors. However, when looking for a SKA package suitable for myself, I refused Mathematica for a simple reason - listing, or how to correctly call it, in general, what is seen on the screen is very similar to the code that is difficult to understand. Here is an example, the Levi-Civita tensor , which, by the way, was named after the famous Italian mathematician Tullio Levi-Chivit , who can say at the origin of the tensor calculus and did a lot for its development. As you know, when working on the General Theory of Relativity, Albert Einstein had to master and make extensive use of tensor calculus. Here are a couple of quotes:
and
I think this explains the fact that at first the tensor calculus was widely used only by theoretical physicists studying the theory of relativity, field theory and gravity, etc. I met records like
Redberry is an open computer algebra system developed and maintained by our compatriots, which in itself is gratifying. However, Redberry is designed to work with abstract tensors (using standard Einstein notation). Such problems usually arise in high energy physics. You can set arbitrary symmetries of tensors, take derivatives, declare functions, etc. Redberry does not allow specifying specific components of the tensor. For example, in Redberry, you can simplify the expression
d f_i (x) / d x_i + g ^ {ab} d f_a (x) / d x_b = 2 d f_i (x) / d x_i
where d f_i (x) / d x_i is the divergence of the vector f_i, and g ^ {ab} is the metric tensor; however, it is not possible to specify specific components of this vector (eg f_0 = x_1 * x_2, f_1 = ..., etc.). Therefore, to bring the calculation to the "numbers" can not. For me, this moment is fundamental. Therefore, alas, we go further.
The Maple software package has been developed by Waterloo Maple since the early 80s. Maple previously contained a tensor package for working with tensors, now this package is included in the package of the DifferentialGeometry Package , working with it at one time really didn’t impress me much. However, much has changed with the advent of the Physics package developed under the guidance of Dr. Edgardo Cheb-Terrab. I note, as I believe, the main advantages:
1. The output format of the calculations, as if we worked with a pencil and paper.
For example:
2. Distinct and convenient Help . For example, the Levi-Civita tensor , compare with the similar reference of the Mathematica package.
3. A platform for communication, exchange of views, questions and answers, etc.
Other SKA for working with tensors, an interested reader can find here .
Thanks for attention.
In comments to this article, ZloAlien asked about mathematical packages for working with tensors, to which the author maisvendoo told him that Maple 18 might be, and attached a picture with the output of tensor package functions. Here, actually, by this I could not pass by.
Computer algebra system (CAS ) computer algebra systems (CAS ) are used for symbolic computations. They are not few, but we will focus on those with whom I had to face, just when looking for the most suitable for myself.
So, in order:
')
Mathematica
Developed in the late 80s by Stephen Wolfram contains, among other things, the Tensors in Physics package for working with tensors. Judging by the description, it can perform basic actions on tensors. However, when looking for a SKA package suitable for myself, I refused Mathematica for a simple reason - listing, or how to correctly call it, in general, what is seen on the screen is very similar to the code that is difficult to understand. Here is an example, the Levi-Civita tensor , which, by the way, was named after the famous Italian mathematician Tullio Levi-Chivit , who can say at the origin of the tensor calculus and did a lot for its development. As you know, when working on the General Theory of Relativity, Albert Einstein had to master and make extensive use of tensor calculus. Here are a couple of quotes:
In one of the letters concerning the new work of Levi-Civita, Einstein wrote: “I admire the elegance of your method of calculation; as it must be pleasant to ride across these fields on a stallion of pure mathematics, while the rest must tirelessly rearrange their legs. ”
and
According to the recollections of D.Ya.Stroyka, when, after Levi-Civita’s death, Einstein was asked what he liked best in Italy, he replied: “Spaghetti and Levi-Civita”
I think this explains the fact that at first the tensor calculus was widely used only by theoretical physicists studying the theory of relativity, field theory and gravity, etc. I met records like
Oh, it's a tensor package, but you don't need it, only theoretical physicists understand it.
Redberry
Redberry is an open computer algebra system developed and maintained by our compatriots, which in itself is gratifying. However, Redberry is designed to work with abstract tensors (using standard Einstein notation). Such problems usually arise in high energy physics. You can set arbitrary symmetries of tensors, take derivatives, declare functions, etc. Redberry does not allow specifying specific components of the tensor. For example, in Redberry, you can simplify the expression
d f_i (x) / d x_i + g ^ {ab} d f_a (x) / d x_b = 2 d f_i (x) / d x_i
where d f_i (x) / d x_i is the divergence of the vector f_i, and g ^ {ab} is the metric tensor; however, it is not possible to specify specific components of this vector (eg f_0 = x_1 * x_2, f_1 = ..., etc.). Therefore, to bring the calculation to the "numbers" can not. For me, this moment is fundamental. Therefore, alas, we go further.
Maple
The Maple software package has been developed by Waterloo Maple since the early 80s. Maple previously contained a tensor package for working with tensors, now this package is included in the package of the DifferentialGeometry Package , working with it at one time really didn’t impress me much. However, much has changed with the advent of the Physics package developed under the guidance of Dr. Edgardo Cheb-Terrab. I note, as I believe, the main advantages:
1. The output format of the calculations, as if we worked with a pencil and paper.
For example:
2. Distinct and convenient Help . For example, the Levi-Civita tensor , compare with the similar reference of the Mathematica package.
3. A platform for communication, exchange of views, questions and answers, etc.
Other SKA for working with tensors, an interested reader can find here .
Thanks for attention.
Source: https://habr.com/ru/post/262287/
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