Divergence, Gradient of higher order tensor

In summary, divergence is a mathematical operation that measures the rate at which a vector field diverges or converges at a given point. It is related to gradient, which is the dot product of the gradient of the vector field. The gradient of a higher order tensor is a tensor that represents the rate of change of the original tensor in all directions, and it is calculated by taking partial derivatives and arranging them in a matrix. These concepts are used in various fields such as fluid dynamics, electromagnetism, and mechanics for understanding and solving complex mathematical problems.
  • #1
chowdhury
36
3
1.) I have the following equation
$$\nabla \cdot \left( \mathbf{A} : \nabla_{s}\mathbf{b} \right) - \frac{\partial^2\mathbf{c}}{\partial t^2} = - \nabla \cdot \left( \mathbf{D}^{Transpose} \cdot \nabla \phi \right )$$

Is my index notation correct?

$$(A_{ijkl} b_{k,l}),_{j} - c_{i,tt} = - (D_{ijk}^{Transpose} \phi_{,k}),j $$
This becomes
$$(A_{ijkl} b_{k,l}),_{j} - c_{i,tt} = - (D_{kij} \phi_{,k}),j $$

2.) New set of A, b, C, d below.
$$\nabla \cdot \left( \mathbf{A} \cdot \nabla \mathbf{b} \right) = \nabla \cdot \left( \mathbf{C} : \nabla_{s}\mathbf{d} \right) $$

Is my index notation correct?
$$(A_{ij} b_{,j}),i = (C_{ijk} d_{j,k})_{,i}$$
 
Last edited:
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  • #2


Yes, your index notation for both equations is correct. In the first equation, you have correctly used the colon notation to represent the double dot product between A and the gradient of b. In the second equation, you have correctly used the comma notation to represent partial derivatives and have also used the colon notation to represent the double dot product between C and the gradient of d. Good job!
 

Related to Divergence, Gradient of higher order tensor

1. What is the concept of divergence?

The divergence of a vector field is a measure of the rate at which the field is spreading out or converging at a certain point. It is represented by the dot product of the gradient operator and the vector field.

2. How is the divergence of a higher order tensor calculated?

The divergence of a higher order tensor is calculated by taking the sum of the partial derivatives of each component of the tensor with respect to its corresponding coordinate axis. This can also be represented using the divergence operator.

3. What does the gradient of a higher order tensor represent?

The gradient of a higher order tensor represents the rate of change of the tensor in each direction. It is a vector field that points in the direction of the greatest increase of the tensor and its magnitude represents the rate of change.

4. How is the gradient of a higher order tensor visualized?

The gradient of a higher order tensor can be visualized using vector field plots or streamlines. These visualizations show the direction and magnitude of the gradient at different points in the tensor field.

5. What are the applications of divergence and gradient of higher order tensors in science?

Divergence and gradient of higher order tensors have various applications in fields such as fluid dynamics, electromagnetism, and mechanics. They are used to model and analyze complex systems and phenomena, such as fluid flow, heat transfer, and stress distribution.

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