For most of the already skilled mathematics and physics workers, this is a very basic problem. But this problem has been bothering me for a long time when I first touched the tensor. So many definitions of tensor, what are the right ones? (Obviously, it's all right.) What are their relationships? I am as simple as I can in my own words to understand the superficial understanding of it.
The concept of tensor was proposed by mathematicians at the end of the 19th century, but the concept really thrived, or after the advent of relativity. The reason is that in theory of relativity, looking at the same physical system under different reference systems, it "looks" differently: for example, the momentum and energy of a particle are associated with different reference systems according to the Lorentz transformation.
This poses a problem: in Bob's opinion, the kinetic amount of a particle is. If you ask Bob how much the particle is moving, he will tell you yes. But after I heard it, I was bound to oppose it: Bob was wrong! The amount of active particles I see is clearly!
We know that Andrew and Bob are right. And can transform each other through the proper Lorentz transformation. "I know what you say," You must have been impatient, "but what is the amount of this particle?" Since the reference system is all affirmative, neither Andrew nor Bob's reference system is superior. Then let's just discard them all. So we say that the kinetic amount of the particle is described by the dynamic tensor. An active tensor is an intrinsic quantity of a physical system that does not change with the coordinates. (If you are confused about the exact meaning of the sentence on the left, please look down first.) It looks in Andrew's coordinate system, and in Bob's coordinate system, it turns into a Lorentz transformation.
You must have found a little feeling right now. What is tensor? As stated in the A.zee book:
a tensor is something that transforms like a tensor! a quantity, in different reference system according to a certain law to transform, is tensor.
What is the advantage of using tensor? The laws of physics do not change with reference systems. Consider one of the following physical processes: two particles 1 and 2 are scattered into 3 and 4. In Andrew's view, the conservation of kinetic quantity is. But this writing does not directly see that Bob also sees the conservation of momentum. But if it was written directly in a tensor language: we immediately knew it was in Andrew's opinion, and in Bob's opinion.
The laws of physics, which are described in tensor language, automatically guarantee that this property does not change with reference system. And from the point of view of the notation, the tensor is more concise. [*]
We have physically understood what a tensor is. Physicists are happy to be here. But the rigorous mathematicians were not satisfied. "You just said tensor is an intrinsic quantity of physical systems that doesn't change with the coordinates," the mathematician questioned, "I know what you say, but what is tensor?"
If one is knows to linear algebra, it is possible to know the concept of linear transformations. The essence of the concept of linear transformations is that it does not depend on the selection of the base of the linear space. Under a set of bases, its matrix representation is a shape; Under another set of bases, its matrix representation is another shape, which is the matrix of the base transformation. There is a common saying:
the meaning of a matrix is a linear transformation, and the similarity matrix is the representation of the same linear transformation under different bases.
Wait a while! "The expression of the same linear transformation under different bases", isn't that the same thing as the tensor that I said before? The Lorentz transformation is the base transformation in the Minkowski space, and the dynamic tensor tensor is essentially a linear transformation. Is the dynamic tensor seen by Andrew and Bob not the representation of this linear transformation under different bases?
You must have found a little feeling right now. What is tensor? In the eyes of mathematicians, tensor has been abstracted into a linear transformation.
Of course, mathematicians can further abstract the concept and extract more common universal property. At this point, the tensor is defined as an element in the tensor product space. Specific definitions are not mentioned here, please refer to the relevant monographs. But despite being abstracted to that extent, the thought behind it remains the same.
If you understand the thought behind tensor by reading above, and then go to the related mathematics or physics monograph or complex or abstract formula, perhaps will be cheerful many:-)
Finally, quoting Mr. Chen Weihuan's "micro-manifold preliminary" a paragraph in the book to summarize:
The
concept of tensor is proposed by G.ricci at the end of 19th century. The purpose of G.ricci study tensor is to find a form that is invariant in coordinate transformation for the expression of geometrical and physical laws. The tensor he considers is an array of vectors, requiring them to obey the law of some linear transformation under the coordinate transformation. Modern theory has described tensor as a multi-linear function of vector space and its dual space, but it is still important to use the component to represent tensor, especially when it comes to tensor calculation.
[*] If the inner product/contraction and other operations are also defined, some invariants can be obtained quickly by the tensor. In this case, the duality space (because the intrinsic product is a linear function) is involved, and then the covariance and contravariance of tensor are involved. For the sake of brevity, I did not mention these concepts in the text. But they are essentially no different from what the body says. Copyright belongs to the author.
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Andrew Shen
Links: http://www.zhihu.com/question/20695804/answer/43265860
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What is tensor (tensor)?
