Divergence of a two dimensional 3 order tensor

In summary, the divergence of a two-dimensional 3 order tensor is a mathematical operation that measures the rate of change of a vector field's magnitude at a given point. It is calculated by taking the dot product of the tensor with the gradient operator, resulting in a scalar value. The applications of this operation are found in fluid dynamics, electromagnetism, continuum mechanics, computer graphics, and image processing. It is different from the curl as it measures expansion or contraction rather than rotation or circulation, and it can be negative, indicating convergence or contraction of the vector field at a given point.
  • #1
MasterD
13
0
I want to calculate the divergence of a two dimensional 3 order tensor; e.g.

nabla=(d/dx, d/dy)

and

Ax =
( C D)
( E F),

Ay =
( G H)
( I J)

(it's a 2x2x2 cube).

Index notation:

(nabla)_i = d/dx_i

and elements of A are A_ijk

How do I contract it properly to calculate

Divergence of A (Nabla inner product A)?

Thanks a lot!
 
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  • #2
Thanks guys.

I got the answer.

You should contract over the first index; so

$(\nabla\cdot\mathbf{A})_{jk}=\frac{d}{dx_i}A{ijk}$
 

1. What is the definition of divergence of a two-dimensional 3 order tensor?

The divergence of a two-dimensional 3 order tensor is a mathematical operation that measures the rate of change of a vector field's magnitude at a given point. It involves taking the dot product of the tensor with the gradient operator, resulting in a scalar value.

2. How is the divergence of a two-dimensional 3 order tensor calculated?

The divergence of a two-dimensional 3 order tensor is calculated by taking the partial derivatives of each component of the tensor with respect to each coordinate axis and then summing them together. This can be expressed mathematically as div(T) = ∂T₁/∂x + ∂T₂/∂y + ∂T₃/∂z.

3. What are the applications of divergence of a two-dimensional 3 order tensor?

The divergence of a two-dimensional 3 order tensor is used in various fields, including fluid dynamics, electromagnetism, and continuum mechanics. It is also applicable in computer graphics and image processing for edge detection and feature extraction.

4. What is the difference between divergence and curl of a two-dimensional 3 order tensor?

The divergence of a two-dimensional 3 order tensor measures the expansion or contraction of a vector field, while the curl measures the rotation or circulation of a vector field. The divergence results in a scalar value, while the curl results in a vector value.

5. Can the divergence of a two-dimensional 3 order tensor be negative?

Yes, the divergence of a two-dimensional 3 order tensor can be negative. It indicates that the vector field is converging or contracting at a given point. A positive divergence indicates that the vector field is expanding or diverging at that point, and a zero divergence indicates that the vector field is neither expanding nor contracting.

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