Vector Calculus: Divergence of Dyadic AB

In summary, the conversation discusses the divergence of a dyadic and the identity for it, which involves taking the laplacian of each component of a vector. The second term of the identity is not a scalar, but rather a result of applying a differential operator to the vector. Writing the equation in components helps clarify this.
  • #1
Mr. Cosmos
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So I have a quick question that will hopefully yield some clarification. So the divergence of a dyadic ##\bf{AB}## can be written as,
$$\nabla \cdot (\textbf{AB}) = (\nabla \cdot \textbf{A}) \textbf{B} + \textbf{A} \cdot (\nabla \textbf{B})$$
where ##\textbf{A} = [a_1, a_2, a_3]## and ##\textbf{B} = [b_1, b_2, b_3]##. However, it seems to me that the first term of the identity yields a vector (the vector ##\bf{B}## scaled by the divergence of ##\bf{A}##), while the second term yields a scalar. Since one cannot simply add a scalar quantity to a vector quantity, it would appear the identity is false, or perhaps I am missing something. I know the dyadic of ##\bf{AB}## yields a tensor, and the divergence of a tensor is a vector, but it doesn't appear that the second term on the right hand side of the identity is a vector. Any help would be greatly appreciated.
 
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  • #2
The second term does not mean a scalar. You are taking the laplacian of each component of ##\mathbf{B}##. The ##j## component of the last term is ##A^{i}\partial_i B^{j}## (using Einstein summation convention). The easiest way to see the identity is to write the equation in components, and it will all make sense - it is just the Leibnitz rule.

Think of ##A^i\partial_i=\mathbf{A}\cdot\mathbf{\nabla}## as a differential operator acting on ##\mathbf{B}## componentwise.
 
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  • #3
Lucas SV said:
The second term does not mean a scalar. You are taking the laplacian of each component of ##\mathbf{B}##. The ##j## component of the last term is ##A^{i}\partial_i B^{j}## (using Einstein summation convention). The easiest way to see the identity is to write the equation in components, and it will all make sense.
Thanks for the quick response. I now see my mistake. Thanks!
 
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Related to Vector Calculus: Divergence of Dyadic AB

1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vector fields and their derivatives, such as gradient, divergence, and curl. It is used to describe and analyze the behavior of physical quantities that have both magnitude and direction, such as forces and velocities.

2. What is the divergence of a dyadic AB?

The divergence of a dyadic AB is a mathematical operation that gives a scalar value representing the rate at which the vector field defined by the dyadic AB is flowing out or in at a particular point. It is given by the dot product of the gradient operator and the dyadic AB.

3. How is the divergence of a dyadic AB calculated?

The divergence of a dyadic AB can be calculated using the formula div(AB) = grad · AB, where grad is the gradient operator and AB is the dyadic AB. This involves taking the partial derivatives of AB with respect to each variable and then computing the dot product of the resulting gradient vector with the dyadic AB.

4. What is the physical significance of the divergence of a dyadic AB?

The physical significance of the divergence of a dyadic AB is that it represents the amount of "source" or "sink" at a particular point in the vector field. A positive divergence means that the vector field is flowing outwards from that point, while a negative divergence means that the vector field is flowing inwards towards that point.

5. How is the divergence of a dyadic AB used in real-life applications?

The divergence of a dyadic AB is used in various fields of science and engineering, such as fluid dynamics, electromagnetism, and heat transfer. It helps in understanding and predicting the behavior of vector fields, which is crucial in designing and optimizing various systems and processes. For example, in fluid dynamics, the divergence of a velocity field is used to determine the presence and strength of sources and sinks, which are important in the study of fluid flow patterns and turbulence.

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