Can the Nabla Operator Be Applied Before Inversion in Tensor Calculations?

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SUMMARY

The discussion centers on the application of the nabla operator in tensor calculations, specifically regarding the divergence of a gradient inverse. It is established that applying the nabla operator to the gradient of a first-order tensor before inverting the resulting second-order tensor is permissible. The inquiry also touches on the interpretation of the inverse gradient, questioning whether it refers to the reciprocal of the nabla operator applied to a scalar field.

PREREQUISITES
  • Understanding of tensor calculus
  • Familiarity with the nabla operator (∇)
  • Knowledge of divergence and gradient operations
  • Basic concepts of inverse operations in mathematics
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  • Learn about the divergence of second-order tensors
  • Explore the implications of tensor inversion in physical applications
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Dear All,

I'm doing some tensor calculation on the divergence of gradient (of a vector) inverse. Am I allowed to first use the nabla operator on gradient and then inverse the whole product?
In other words, I'm searching for the divergence of a 2nd order tensor which is itself inverse of (gradient of a first order tensor ). Thanks in advance
 
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What do you mean the inverse gradient?

Is it something like \frac{1}{\nabla} \phi?
 

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