Correct differentiation identity? (tensors, vectors)

The conversation is about a mathematical problem involving the differentiation identities for vectors, scalars, and tensors. The person is looking for sources to confirm their understanding and has found differentiation identities involving vectors and scalars or tensors, but not a combination of both. They then provide a specific identity using notation for vectors, tensors, and the double dot product, and ask if it is correct. The expert responds by summarizing that the provided identity is correct and provides a source for more information in a common notation. In summary, the conversation is about a mathematical problem involving differentiation identities and the person has found one involving vectors and scalars or tensors, but not a combination of both. The provided identity is correct and a source is provided for more information
  • #1
Waldheri
6
0
Hello,

I'm working on some problems and I want to pose the following, though I am not completely sure it is correct. Can somebody point me to some sources on this? I have tried googling myself, but I only found differentiation identities with either just vectors and scalars on the on hand, or dot products between tensors on the other hand.

Note: [itex]\boldsymbol{A}[/itex] denotes a second-order tensor, [itex]\vec{v}[/itex] denotes a vector and [itex]:[/itex] the double dot product.

[tex]\nabla\cdot(\vec{v}\cdot\boldsymbol{A}) = \nabla\vec{v}:\boldsymbol{A} + \vec{v}\cdot\nabla\cdot\boldsymbol{A}[/tex]

So in other words: is the above identity correct?
 
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Mathematics news on Phys.org

1. What is a tensor?

A tensor is a mathematical object that can represent a linear relationship between different sets of data. It is a generalization of vectors and matrices, and can have multiple components in different directions.

2. How is a tensor different from a vector?

A vector is a one-dimensional object that has both magnitude and direction. In contrast, a tensor can have multiple dimensions and components, and does not necessarily have a specific direction.

3. What is the correct way to differentiate a tensor?

The correct way to differentiate a tensor is to use the product rule, chain rule, and partial differentiation. This involves taking the partial derivative of each component of the tensor with respect to the corresponding variable.

4. Can tensors be used to represent physical quantities?

Yes, tensors can be used to represent physical quantities such as velocity, force, and stress. They are commonly used in physics and engineering to describe the relationships between different physical quantities.

5. What are some applications of tensors in science?

Tensors have many applications in science, including in physics, engineering, and computer science. They are used to model physical phenomena, solve problems in mechanics and electromagnetism, and perform calculations in machine learning and image processing.

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