Divergence of vector/tensor

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In summary, the conversation discusses the use of the divergence theorem to manipulate an integral involving a symmetric rank 2 tensor and a vector. The speaker suggests splitting the integral and using the divergence theorem to prove its equality.
  • #1
Qyzren
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Hi guys, trying to solve a problem in MHD, i realized i need to be able to take the divergence of this following integral, but I don't know how to do it.
M is a symmetric rank 2 tensor, r is a vector.

The integral is as follows
[tex]\int_{\partial V} (\textbf{r} d \textbf{S} \cdot \textbf{M}+d\textbf{S} \cdot \textbf{Mr})[/tex]
I need to somehow manipulate this to get [tex]\int_V {\{\nabla \cdot \textbf{M})\textbf{r}+\textbf{r}(\nabla \cdot \textbf{M})+2\textbf{M}\}dV}[/tex]

Thanks
 
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  • #2
The divergence theorem states that the divergence of a tensor or vector field over a volume V is equivalent to a surface integral of the inner product of the tensor or vector field with the surface basis vectors. So split the integral into the sum of two integrals and use the divergence theorem to prove the equality.
 

1. What is the definition of divergence of a vector/tensor?

The divergence of a vector or tensor is a mathematical operation that measures the rate at which the vector or tensor is spreading out or converging at a given point in space. It is represented by the dot product of the vector/tensor with the del operator (represented by ∇).

2. How is divergence calculated?

The divergence of a vector/tensor is calculated by taking the dot product of the vector/tensor with the del operator (∇) and then taking the sum of the resulting components. This can also be represented by the divergence operator (represented by ∇ ·) acting on the vector/tensor.

3. What is the physical significance of divergence?

The physical significance of divergence is that it represents the flow or flux of a vector/tensor field through a given surface or volume. A positive divergence indicates a net outward flow, while a negative divergence indicates a net inward flow. In fluid dynamics, for example, the divergence of a velocity field is related to the rate of expansion or compression of the fluid at a given point.

4. What are the properties of divergence?

Some important properties of divergence include linearity (i.e. the divergence of a sum of vectors/tensors is equal to the sum of their individual divergences), and the product rule (i.e. the divergence of a product of a scalar and a vector/tensor is equal to the product of the scalar and the divergence of the vector/tensor). It is also invariant under proper rotations and translations.

5. How is divergence used in real-world applications?

Divergence is used in various fields of physics and engineering, such as fluid dynamics, electromagnetism, and continuum mechanics. It is used to describe the behavior of flow fields, electric and magnetic fields, and stress/strain fields in materials. In practical applications, it can be used to analyze and model the behavior of fluids, design electrical circuits, and predict the structural integrity of materials.

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