What are the conditions for the divergence of a function of r to be true?

In summary, the statement \vec \bigtriangledown \cdot \vec f ( \vec r ) = \frac {\partial}{\partial r} (r^2 | \vec f ( \vec r ) | ) is not always true and requires certain conditions on \mathbf{f}(\mathbf{r}) to hold. For example, if \mathbf{f}(\mathbf{r}) = \hat{\mathbf{x}}, the divergence is zero and the magnitude is one, resulting in \frac{\partial}{\partial r}(r^2 |\mathbf{f}(\mathbf{r})|) = 2r, making it more of a differential equation than an identity.
  • #1
avidfury
1
0
Why is this true?

[tex]
\vec \bigtriangledown \cdot \vec f ( \vec r ) = \frac {\partial}{\partial r} (r^2 | \vec f ( \vec r ) | )
[/tex]
 
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  • #2
It's not, at least not without some conditions on [itex]\mathbf{f}(\mathbf{r})[/itex] that you haven't given us. Take [itex]\mathbf{f}(\mathbf{r}) = \hat{\mathbf{x}}[/itex], for example. The divergence is zero, and the magnitude is just one, so

[tex]\frac{\partial}{\partial r}(r^2 |\mathbf{f}(\mathbf{r})|) = \frac{\partial}{\partial r}(r^2) = 2r[/tex]

[itex]\nabla \cdot \mathbf{f}(\mathbf{r}) = \partial_r(r^2|\mathbf{f}(\mathbf{r})|)[/itex] looks more like a differential equation to be solved rather than an identity.
 

What is the divergence of a function of r?

The divergence of a function of r is a mathematical concept that measures how much the vector field associated with that function is spreading out or converging at a given point.

How is the divergence of a function of r calculated?

The divergence of a function of r is calculated using the gradient and dot product. Specifically, it is the dot product of the gradient and the vector field associated with the function.

What does a positive/negative divergence indicate?

A positive divergence indicates that the vector field is spreading out at a given point, while a negative divergence indicates that the vector field is converging at that point.

What are some real-world applications of the divergence of a function of r?

The divergence of a function of r has many applications in physics and engineering, such as in fluid dynamics to study the flow of liquids and gases, in electromagnetism to analyze electric and magnetic fields, and in thermodynamics to study heat transfer.

Can the divergence of a function of r be visualized?

Yes, the divergence of a function of r can be visualized using vector fields on a graph. Arrows pointing away from a point indicate a positive divergence, while arrows pointing towards a point indicate a negative divergence.

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