What book is highly recommended as a tensor textbook?

[good_phy]

Hi.

i'm actually under the department of the applied optics so I had very few change to

face the concept, tensor.

But my research topics is highly related to particle acceleator so tensor concept is need to

be understanded to go straightfoward.


I'm looking for tensor textbook which include enough page and easy explanation for tensor

concept and its application for special relativity and even introduction level of general

relativity. Too much rigorous methematical approach need to be avoided. I'm student under

applied physics, not theorectical physicis.


Please guide me.

 

[bcrowell]

Most physicists learn about tensors from a GR book. If you want a GR book, you could use an easy undergraduate book like Hartle.

 

[Daverz]

Though it's presented as a graduate text, I think the opening chapters of Sean Carroll's book are pretty accessible.

Also, the Schaum's Outline isn't bad.

 

[jasonRF]

An excellent basic book on tensors is "a brief on tensor analysis" by Simmonds. Doesn't cover relativity, though.

jason

 

[SinghRP]

I recommend:
An Introduction to Riemannian Geometry and the Tensor Calculus, C.E. Weatherburn. Cambridge, At The University Press, 1963.

 

[mathwonk]

Although mathematical, I suggest looking at chapter 4 volume I of Spivak's Comprehensive introduction to differential geometry. There he gives a useful dictionary relating the classical with the modern notation and terminology for tensors. This is helpful when trying to pass between different treatments of the subject. If one learns only from a classical book, one is helpless when trying to read modern books. This reference is only for the mathematics of tensors, as there is no discussion there of their use in physics.

 

[SinghRP]

I recommend also Applied Mathematics for engineers and Physicists, Louis A. Pipes, McGraw-Hill Book. This one covsrs physical meanings of contravariant vectors, covariant vectors, and tensors.
Alternatively, a book on the 'physics of continua' would have excellent physical meanings of tensors.

I have not noticed changes in tensor notations.

 

[mathwonk]

Recall that a vector space, such as a tangent space to a surface, has algebraic structures of addition and scalar multiplication only. These are useful for representing velocities of particles. Tensors are structures on a vector space that involve also multiplication of some type. thus a dot product is a tensor, as is also a determinant. They are used for measuring angles, lengths, areas, volumes, as well as curvature of surfaces obtained by multiplying curvatures of two curves together, and presumably many other physical concepts.

Thus mathematically, tensors are merely forms of multiplication of more than one vector. Hence they are defined in ways like this: a 2- tensor is a multiplication of pairs of vectors, i.e. a function VxV-->R which acts like a multiplication, i.e. is linear in each variable separately, or "multilinear". E.g. if V = R^2, the determinant of the matrix with rows (u,w) gives a multilinear map VxV––>R that is also alternating. Other tensors like the dot product are symmetric in their variables. Thus important special types of tensors are the symmetric ones or the alternating ones. The alternating ones are closely related to differential forms.

As to notation, the "classical" notation (used by 19th century mathematicians and adopted in the early 20th century by Einstein) is heavy with upper and lower indices, i.e. it focuses on the shape of the symbols used as coefficients. The "modern" notation (used for the last 75 years or so by mathematicians) emphasizes more the algebraic properties of the tensors over the symbols used to represent them.