Divergence Simplification/Identities

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In summary, the conversation is about simplifying the equation \nabla\bullet (a \bullet b)b and if it can be written in any other way. The participants discuss using the general equation \nabla (\varphi \mathbf{F})=(\nabla \varphi)\bullet\mathbf{F}+\varphi (\nabla \bullet \mathbf{F}) and substituting in a and b for \varphi and \mathbf{F}. The conversation also mentions using the gradient of vector dot product and the Levi Civita tensor to simplify the equation further, but notes that it may still look ugly.
  • #1
feedmeister
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Quick question…

what does the following simplify to? Can it be written in any other way?

[itex]\nabla\bullet (a \bullet b)b [/itex]

where a and b are vectors.

Thanks,
 
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  • #2
feedmeister said:
Quick question…

what does the following simplify to? Can it be written in any other way?

[itex]\nabla\bullet (a \bullet b)b [/itex]

where a and b are vectors.

Thanks,

in general

[tex]\nabla (\varphi \mathbf{F})=(\nabla \varphi)\bullet\mathbf{F}+\varphi (\nabla \bullet \mathbf{F})[/tex]

let [tex]\varphi = \mathbf{a}\bullet \mathbf{b}[/tex]

and [tex]\mathbf{F}=\mathbf{b}[/tex]
 
  • #3
IssacNewton said:
in general

[tex]\nabla (\varphi \mathbf{F})=(\nabla \varphi)\bullet\mathbf{F}+\varphi (\nabla \bullet \mathbf{F})[/tex]

let [tex]\varphi = \mathbf{a}\bullet \mathbf{b}[/tex]

and [tex]\mathbf{F}=\mathbf{b}[/tex]

Thanks, but I didn't think that [itex]\nabla\bullet (\mathbf{a} \bullet \mathbf{b})\mathbf{b} [/itex] was the same as [tex]\nabla (\varphi \mathbf{F})[/tex]... there's still a [tex]\bullet[/tex] between the [tex]\nabla [/tex] and the rest of the statement.

Can you clarify?
 
  • #4
IssacNewton said:
in general

[tex]\nabla (\varphi \mathbf{F})=(\nabla \varphi)\bullet\mathbf{F}+\varphi (\nabla \bullet \mathbf{F})[/tex]

little mishtake...above should be

[tex]\nabla \bullet (\varphi \mathbf{F})=(\nabla \varphi)\bullet\mathbf{F}+\varphi (\nabla \bullet \mathbf{F})[/tex]

:-p
 
  • #5
Thanks, IssacNewton.

When substituting in [tex]\mathbf{a}\bullet \mathbf{b}[/tex] and [tex]\mathbf{b}[/tex] into the equation, it looks like it'd simplifies further.. but it looks like it'd be ugly.

Any good way of simplifying it?

Thanks,
 
  • #6

Related to Divergence Simplification/Identities

1. What is divergence simplification?

Divergence simplification is a mathematical technique used to simplify complex expressions involving divergence, which is a measure of the flow of a vector field from a given point. It involves applying various identities and rules to reduce the complexity of the expression and make it easier to work with.

2. Why is divergence simplification important?

Divergence simplification is important because it allows us to solve problems and analyze systems that involve vector fields more easily. By simplifying the expressions, we can gain a better understanding of the underlying concepts and make accurate predictions about the behavior of the system.

3. What are some common identities used in divergence simplification?

Some common identities used in divergence simplification include the product rule, quotient rule, and chain rule. Other identities such as the divergence theorem and Stokes' theorem can also be used in certain situations.

4. How do you apply divergence simplification in practice?

To apply divergence simplification, you first need to identify the vector field and its components in the given expression. Then, you can use the identities and rules to manipulate the expression and simplify it. It is important to keep track of any changes made and make sure the final expression is equivalent to the original one.

5. Are there any limitations to divergence simplification?

While divergence simplification can be a powerful tool, it does have its limitations. It may not always be possible to simplify an expression using existing identities, especially if the vector field is complex. In these cases, other techniques such as numerical methods may be necessary to solve the problem.

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