Divergence of Gradient inverse

In summary, the divergence of gradient inverse is a mathematical concept used to describe the rate of change of a vector field. It is calculated by taking the dot product of the gradient of a vector field with the inverse of that same gradient. It has various applications in physics, engineering, and mathematics, and is closely related to the fundamental theorem of calculus. Real-world examples include analyzing fluid flow, electric fields, and heat transfer.
  • #1
Compengineering
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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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  • #2
What do you mean the inverse gradient?

Is it something like [tex] \frac{1}{\nabla} \phi [/tex]?
 

1. What is the definition of "Divergence of Gradient inverse"?

The divergence of gradient inverse is a mathematical concept that describes the rate of change of a vector field in terms of its sources (e.g. charges, mass) and sinks (e.g. sinks, drains). It is represented by the symbol ∇ · (∇-1).

2. How is the "Divergence of Gradient inverse" calculated?

The divergence of gradient inverse is calculated by taking the dot product of the gradient of a vector field with the inverse of the gradient of that same vector field. This can also be expressed as the Laplacian operator (∇2) acting on the vector field.

3. What are the applications of "Divergence of Gradient inverse" in science?

The divergence of gradient inverse has various applications in physics, engineering, and mathematics. It is used to solve differential equations, analyze fluid flow, and study electromagnetic fields, among other things.

4. What is the relationship between "Divergence of Gradient inverse" and the fundamental theorem of calculus?

The fundamental theorem of calculus states that the integral of a function over an interval can be evaluated by finding an antiderivative of the function and evaluating it at the endpoints of the interval. The divergence of gradient inverse is closely related to this theorem, as it is essentially a generalization of the fundamental theorem to vector fields.

5. Are there any real-world examples that demonstrate the concept of "Divergence of Gradient inverse"?

Yes, there are many real-world examples that involve the concept of divergence of gradient inverse. For instance, in fluid dynamics, it is used to analyze the flow of a fluid through a pipe with varying cross-sectional area. It is also used in electrostatics to calculate the electric field around a point charge, and in heat transfer to study the flow of heat through a material with varying thermal conductivity.

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