Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

When defining tensors as multilinear maps, I am having trouble understanding why a tensor, let's say of type (2,1), can be written in the following way:

$$T = T^{\mu\nu}_{\rho} e_\mu \otimes e_\nu \otimes f^\rho$$

where $\{e_\mu\}$ is a basis for the vector space, and $\{f^\mu\}$ is its dual basis.

I know how to expand the tensor when applied to some generic vectors and covectors. So in the example above:

$$T(\alpha,\beta,X) = T(\alpha_\mu f^\mu,\beta_\nu f^\nu, X^\rho e_\rho) = \alpha_\mu \beta_\nu X^\rho T(f^\mu,f^\nu,e_\rho) = \alpha_\mu \beta_\nu X^\rho\, T^{\mu\nu}_{\rho}$$

How would I proceed from here? Can I just say $\alpha_\mu = \alpha_\nu f^\nu(e_\mu) = \alpha (e_\mu) $, and similarly for the others? Then I would get:

$$T(\alpha,\beta,X) = T^{\mu\nu}_{\rho}\, \alpha(e_\mu)\, \beta(e_\nu)\, f^\rho(X)$$

I am not sure how to generalise from here. Apologies if this is really obvious.

share|cite|improve this question
up vote 2 down vote accepted

You have the right idea--by the time you're at your last step, everything is just numbers. At this point, just einstein sum your indices.

I will say that most working people, unless they are deriving general results, will start from the tensor in an explicit coordinate basis, and infer the mathematical object. This way is more formally "correct", but a lot less useful for actually figuring out physics, exactly due to all of these gymnastics.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.