# Intuitively understanding div(curl F) = 0

• rolfz
In summary, the conversation discusses the concept of the identity div(curl F) = 0 and provides a physical explanation for this property by using Gauss's and Stokes's Theorems. It also mentions the relationship between the curl and divergence operators and how they measure different types of variations in a vector field. The conversation ends with a thank you to those who provided helpful responses.
rolfz
Hello all!

I have been reviewing my vector calculus coursework as of late, and this time around, I've been really trying to understand the concepts intuitively/visually instead of just the math. Unfortunately, the identity div(curl F)=0 is giving me trouble.

I understand divergence is a measure of a vector field's compressibility. I understand curl is a vector field, representing F's rate of rotation. What I'm having a hard time visualizing is why curl F always produces a vector field that is incompressible?

If you know about exterior derivatives, then this identity is equivalent to $d^2 = 0$.

To give a "physical picture" for this identity, first use Gauss's Theorem for $\mathrm{div}(\mathrm{curl} \, \vec{F})$ to get:
$$\int_{\Omega}{d^3x \, \mathrm{div}(\mathrm{curl} \, \mathrm{F})} = \oint_{\Sigma}{da \, \hat{n} \cdot (\mathrm{curl} \, \vec{F})}$$
where $\Omega$ is an arbitrary volume element bounded by the closed surface $\Sigma$, with an exterior unit normal $\hat{n}$ everywhere.

Next, cut the boundary surface $\Sigma$ in two by a plane. Let the intersection of the plane with the boundary surface be a curve C, and let the two halves of the boundary surface be $\Sigma_1$, and $\Sigma_2$. We would have:
$$\oint_{\Sigma}{da \, \hat{n} \cdot (\mathrm{curl} \, \vec{F})} = \iint_{\Sigma_1}{da \, \hat{n} \cdot (\mathrm{curl} \, \vec{F})} + \iint_{\Sigma_2}{da \, \hat{n} \cdot (\mathrm{curl} \, \vec{F})}.$$
Now, we can apply Stokes's Theorem for each of these surface integrals. However, notice that if we go along the curve in the clockwise direction, then, according to the right-hand rule, the normal to the surface would always be exterior to one of the surfaces, whereas it would be interior to the other of the surface $\Sigma_1$, and $\Sigma_2$. Since $\hat{n}$ is the exterior normal on both of them, we need to circumvent the loop C in opposite directions, and the integrals become:
$$\iint_{\Sigma_1}{da \, \hat{n} \cdot (\mathrm{curl} \, \vec{F})} + \iint_{\Sigma_2}{da \, \hat{n} \cdot (\mathrm{curl} \, \vec{F})} = \oint_{C^{+}}{d\vec{l} \cdot \vec{F}} + \oint_{C^{-}}{d\vec{l} \cdot \vec{F}} = 0$$
since changing the direction of circumvention in a line integral of the second kind changes the sign in front of it.

Take a simple but nontrivial vector field with a nonconstant curl.
$F = \hat{x} y^2$.

The curl of this is $2y\hat{z}$.

This tells you that the curl in this case points in a direction which is perpendicular to the direction of the original vector and perpendicular to the direction containing the spatial variation. (this makes sense if you think of the curl as "nabla cross F")

The divergence operator on the other hand is measuring variation along the unit vector; how the z component varies along z, and so on.

The divergence of $2y\hat{z}$ is zero because the variation of the z component of the vector is zero along the z direction.

This is not rigorous of course but should give you a sense of the "perpendicularity" of "nabla dot curl".

Last edited:
Antiphon said:
Take a simple but nontrivial vector field with a nonconstant curl.
$F = \hat{x} y^2$.

The curl of this is $2y\hat{z}$.

This tells you that the curl in this case points in a direction which is perpendicular to the direction of the original vector and perpendicular to the direction containing the spatial variation. (this makes sense if you think of the curl as "nabla cross F")

The divergence operator on the other hand is measuring variation along the unit vector; how the z component varies along z, and so on.

The divergence of $2y\hat{z}$ is zero because the variation of the z component of the vector is zero along the z direction.

This is not rigorous of course but should give you a sense of the "perpendicularity" of "nabla dot curl".

Dickfore and Antiphon, thank you both so much for your thoughtful replies.

Antiphon, this was precisely what I was looking for! My gut told me something like this was what was going on when I took the curl of a vector field, but I've always had trouble visualizing things in 3D.

Thank you both once again!

Erebus_Oneiros
Or, even more imprecisely:
The curl tells you how a vector "moves" WITHIN a tiny disk (i.e local rotation in some specified plane the disk lies in), whereas the divergence tells how the vector "moves" ACROSS the disk.

## 1. What does "div(curl F) = 0" mean?

"div(curl F) = 0" is a mathematical expression that represents the divergence of the curl of a vector field. In simpler terms, it means that the flow of a vector field is equal to 0, indicating that the vector field is not spreading out or converging at any point.

## 2. Why is it important to understand div(curl F) = 0?

Understanding div(curl F) = 0 is important because it helps us understand the behavior of vector fields and their flow. It is also a fundamental concept in vector calculus and is used in various fields such as physics, engineering, and fluid dynamics.

## 3. How can I intuitively understand div(curl F) = 0?

One way to intuitively understand div(curl F) = 0 is to think of the vector field as representing the flow of a fluid. If the fluid is not spreading out or converging at any point, then the divergence of the curl must be equal to 0. Another way is to visualize the vector field as a series of closed loops, where the curl represents the circulation around each loop and the divergence represents the net flow of the vector field.

## 4. What is the relationship between div(curl F) = 0 and the conservation of mass?

There is a direct relationship between div(curl F) = 0 and the conservation of mass. In physics, the divergence of a vector field represents the net flow of a quantity, such as mass or energy. Therefore, if the divergence of the curl is equal to 0, it means that the net flow of the vector field is 0, which aligns with the principle of conservation of mass.

## 5. How is div(curl F) = 0 related to the Helmholtz decomposition theorem?

The Helmholtz decomposition theorem states that any vector field can be decomposed into two components: a divergence-free component and a curl-free component. In other words, any vector field can be represented as the sum of a potential field (represented by the gradient of a scalar function) and a solenoidal field (represented by the curl of a vector function). Therefore, when div(curl F) = 0, it means that the vector field can be expressed solely as a potential field, which is a key component of the Helmholtz decomposition theorem.

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