Index Notation and Dual Vectors

In summary, to find the dual vector of the given tensor, we use the equation dj= EijkTik where Eijk is defined as 1, 0, or -1 depending on certain conditions. By plugging in the values of the tensor and solving for dj for j=1,2,3, we can obtain the dual vector.
  • #1
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Homework Statement


Find the dual vector of the following tensor:

6 3 1
4 0 5
1 3 2


Homework Equations



dj= EijkTik

Where Eijk = 1 if ijk=123, 231, 312
Eijk = 0 if i=j i=k or j=k
Eijk = -1 if ijk = 132, 213, 321

The Attempt at a Solution



Ok, so I'm not really sure how to solve this but my thought is that it is simply multiply diagonally across the matrix such that

dj= (6*0*2)*(1) + (3*5*1)(1) +(1*4*3)*(1) + (1*0*1)(-1) + (6*5*3)(-1) +(3*4*2)(-1)

However, I don't see how that gets me a vector in the end.

Please HELP!
 
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  • #2
dj is the jth component of the dual vector. So there are three different values d1, d2 and d3. Just put j=1,2,3 and find them.
 

1. What is index notation?

Index notation is a mathematical notation used to represent vectors and tensors in a concise and convenient manner. It uses subscripts and superscripts to represent the components of a vector or tensor, making it easier to perform calculations and manipulate equations.

2. How is index notation used in physics and engineering?

Index notation is commonly used in physics and engineering to represent physical quantities such as forces, velocities, and stresses. It allows for efficient calculation and manipulation of equations, making it a useful tool in these fields.

3. What are dual vectors?

Dual vectors, also known as covectors, are mathematical objects that represent linear functionals on a vector space. In other words, they are vectors that can act on other vectors to produce a scalar value.

4. How are dual vectors related to index notation?

In index notation, dual vectors are represented by superscripts, while vectors are represented by subscripts. This notation allows for the easy identification and manipulation of dual vectors in equations and calculations.

5. Can index notation be used in higher dimensions?

Yes, index notation can be extended to higher dimensions, such as three-dimensional space. In this case, additional subscripts and superscripts are used to represent the extra dimensions. This allows for the representation and manipulation of more complex vectors and tensors.

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