Divergence, Gradient of higher order tensor

[chowdhury]

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?
$$(A_{ij} b_{,j}),i = (C_{ijk} d_{j,k})_{,i}$$

 

[jvicens]

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!