# Tensors: Exploring Indices, Equations, & Transformations

• DreamComeTrue
In summary, the equations in (a) conform to tensor notation, while the equations in (b) do not. The equations in (c) follow the rules for a contravariant and covariant vector, and for a second-rank contravariant tensor.

#### DreamComeTrue

1. (a) Remembering the distinction between summation indices and free indices, look at the following equations and state whether they conform to tensor notation, and if not why not:
(i) Tmn=Am^nB
(ii) Uij^i=Ai^kDk
(iii) Vjk^ii=Ajk
(iv) Ai^j=Xi^iC^j+Yi^j

(b) (i) Write out in full the equations bi=djgi^j (d-differential) in a 2-dimensional space.
(ii) If g^ij is the inverse of the metric tensor gij and di=d/dx^i, what are the values of the components of bi=djgi^j ?

(c) For this part, you should use the tensor transformation rules for a contravariant and covariant vector, and for a second-rank contravariant tensor:
P^i=(dx^i/dx^a)P^a, Pi=(dx^a/dx^i)Pa, T^ij=(dx^i/dx^a)(dx^j/dx^b)T^ab
, , .

(i) If A^i and B^j are contravariant vectors, prove that transforms as A^iB^j a contravariant second-rank tensor.

(ii) If A^i is a contravariant vector and Ci is a covariant vector, prove that A^iCi is a scalar field.

(iii) If T^ij is a skew-symmetric contravariant second-rank tensor, prove that its skew symmetry property is invariant under tensor transformations.

Anyone who can help me with that?

Many thanks

Mary

Firstly, we don't do people's homework for them here. You must show your work before we can help you.

Secondly, your notation is ambiguous: it is common to denote subscripts with underscores (A_n), superscripts with hats, or whatever they're called (A^n), and multiple indices in brackets (A^{mn}).

Welcome to PF!

Hi Mary! Welcome to PF!

Hi

Thanks for ur response:)

I attached the relevant file, hope that now it's easier

Thanks again

MAry:)

#### Attachments

• TENSOR.doc
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Hi Mary! Thanks for the PM.

Sorry, I don't like .doc files …

I was hoping someone else would anwser …

can't you type it?

While the .doc file is certainly easier to read...
1) It would be nice for you to re-post here, as many people don't like .doc files, and as Tiny-Tim said there are X^2 and X_2 tags in the reply box.
2) As Cristo said, we don't do peoples' homework here. Sure we can help you with it... but please post an attempt at the problem, or indicate specifically what is giving you problems, so that we may have an easier time helping without doing all of your work for you.

First of all, can you tell us what conditions need to be satisfied in a proper tensor algebraic expression?

## 1. What are tensors?

Tensors are mathematical objects used to represent and manipulate multidimensional data. They have a defined set of indices that determine how they transform under coordinate transformations.

## 2. What is the significance of indices in tensors?

Indices in tensors represent the dimensions of the data being represented. They allow for the manipulation and transformation of the data in a coordinate-independent manner.

## 3. How are tensors used in physics?

Tensors are used in physics to represent physical quantities that have both magnitude and direction, such as force and velocity. They also play a crucial role in the equations of general relativity.

## 4. What are the different types of transformations that tensors can undergo?

Tensors can undergo different types of transformations, such as rotation, translation, and scaling. These transformations can be represented mathematically using tensor equations.

## 5. How are tensors used in machine learning and data analysis?

Tensors are used in machine learning and data analysis to represent and manipulate multidimensional data, such as images and text. They allow for efficient data processing and analysis, making them a valuable tool in these fields.