# Math Methods for Physicists by Arfken Questions?

• Abolaban
In summary, the conversation discusses the use of equations 2.66 in transforming from 4-D to 11-D space in tensor analysis. It is mentioned that the rank and dimension of a tensor are different concepts and that the dimension is implicit in the values of the indices.
Abolaban
Hello Big Minds,

I am reading through Math Methods for Physicists by Arfken and Weber 6th ed --Tensor Analysis.

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Note1:
In page 133, in the foot note it is said that "In N-dimensional space a tensor of rank n has N^n componenets"...in page 135 I understood from the written text that Contravariant, Covariant and Mixed tensors set of equations (2.66) works independent of reference frame and space dimension.
Q1: how can one use equations 2.66 to transform from 4-D (Minkowiski space-time) )to 11-D dimension (as one proposal of String theory) for instance while both sides of Tensors in equations 2.66 must have the same rank (this case of rank 2)? Simply speaking, the number of Tensor's indices depend on the rank of the tensor and the rank of the tensor depends on the dimension but equations 2.66 are independent of the dimension...how is that?
plus his indices of coordinate transformation of contravariant were defined in the subscript first 2.62a but they apear in the superscript in the first equation of 2.66...why? (they must be the same as in this link)

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Abolaban

Last edited:
I think you're confusing rank with dimension.

A tensor of rank 2 is like a matrix. In 3 dimensions the matrix would be 3x3, for 4 dimensions 4x4 and for 11 dimensions it would be 11x11.

The dimension comes into play in a tensor in the value of the indices ie I and j values for 3 dimensions would be { 1, 2, 3 } and for 11 would be { 1, 2, 3, ... 11 }

do you mean that equations 2.66 depend implicitly on dimensions?

I guess you could say that. It's understood by the reader what values the indices may take and so if you as the author are stating these equations then you would've also said the dimension of the space in which they are valid.

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The purpose of studying Math Methods for Physicists by Arfken is to provide a comprehensive understanding of the mathematical tools and techniques that are essential for solving problems in physics. These methods include vector calculus, complex analysis, differential equations, and linear algebra, among others.

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This book is generally recommended for intermediate to advanced level physics students, as it assumes a basic understanding of calculus and physics concepts. However, beginners may also benefit from the book if they have a strong foundation in mathematics and are willing to put in the effort to understand the material.

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One of the key differences of this book is its comprehensive coverage of a wide range of mathematical techniques, including more advanced topics like group theory and special functions. It also includes numerous example problems and exercises with solutions, making it a valuable resource for self-study.

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Yes, this book is often used as a reference guide by physicists and other scientists due to its clear explanations and extensive coverage of mathematical methods. It is also organized in a way that makes it easy to find specific topics and techniques when needed.

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