Tensor of Type (k,l): John Lee, Wald's GR

In summary, the conversation discusses the definition of a tensor of type (k,l) in John Lee's books "Introduction to smooth manifolds" and "Riemannian manifolds" and in Wald's "General relativity". The definition states that a tensor of type (k,l) can be represented as a member of a certain vector space or as a multilinear function. However, there is a difference in notation between math and physics, with one using (k,l) and the other using (l,k). It is unclear if this is a "math vs. physics" difference or if one of the parties is not following the standard convention.
  • #1
Fredrik
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In John Lee's books "Introduction to smooth manifolds" and "Riemannian manifolds", a tensor of type [tex]\begin{pmatrix}k\\ l\end{pmatrix}[/tex] on a vector space V is defined as a member of

[tex]\underbrace{V^*\otimes\cdots\otimes V^*}_{k}\otimes\underbrace{V\otimes\cdots\otimes V}_{l}[/tex]

or as a multilinear function

[tex]\underbrace{V^*\times\cdots\times V^*}_{l}\times\underbrace{V\times\cdots\times V}_{k}\rightarrow\mathbb R[/tex]

or, when [itex]l>0[/itex], as a multilinear function

[tex]\underbrace{V^*\times\cdots\times V^*}_{l-1}\times\underbrace{V\times\cdots\times V}_{k}\rightarrow V[/tex]

(These three vector spaces are isomorphic). But in Wald's "General relativity", this is called a tensor of type [itex](l,k)[/itex]. I just want to ask, is this a "math vs. physics" thing, like when physicsts make their inner products antilinear in the first variable and mathematicians make theirs antilinear in the second? Or is there a standard convention that one of these guys is ignoring?
 
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  • #2
i saw the type of (l,k) for both of math and physics.
 

1. What is a Tensor of Type (k,l)?

A Tensor of Type (k,l) refers to a mathematical object that represents a linear transformation between vector spaces. The 'k' and 'l' represent the number of contravariant and covariant indices, respectively. This type of tensor is commonly used in the study of General Relativity.

2. Who is John Lee?

John Lee is a renowned mathematician and author who specializes in differential geometry and topology. He is the author of the book "Introduction to Smooth Manifolds" which is a widely used textbook in graduate-level mathematics courses.

3. What is Wald's GR?

Wald's GR refers to the theory of General Relativity as developed by Robert Wald, a theoretical physicist and professor at the University of Chicago. His book "General Relativity" is a standard reference for graduate-level courses on the subject.

4. How is Tensor of Type (k,l) used in General Relativity?

Tensors of Type (k,l) are used extensively in General Relativity to describe the curvature of spacetime. They are essential in formulating Einstein's field equations, which describe how matter and energy affect the geometry of the universe.

5. Are there any real-life applications of Tensor of Type (k,l) in science?

Yes, Tensors of Type (k,l) have many real-life applications in science, particularly in the fields of physics and engineering. They are used in a variety of areas such as fluid dynamics, electromagnetism, and quantum mechanics. In addition, they are essential in computer graphics and image processing for tasks such as image recognition and object tracking.

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