In layman's terms, what are curl and divergence?

Question

As the title says, I'm wondering what $curl(\Bbb F)$ and $div(\Bbb F)$ mean, assuming $\Bbb F$ is a vector force field. Today in class I learned that if $\Bbb F$ is conservative, $curl(\Bbb F) = \vec 0$, and we got a generalized reason for why it is referred to as conservative (having to do with the conservation of energy). But what is curl? I heard from someone that it had to do with how "swirly" the function is, but they were unsure as well. Also is divergence just the opposite of curl? Thanks in advance!

Answer 1

You can visualize these concepts very well in fluid dynamics. Given a vector field $\mathbf F$, you can interpret it it as a velocity field of some fluid. Its divergence $\mathrm{div}(\mathbf F)$ at a certain point $P$ represents the total outgoing flux across a small closed surface (shrinking to $P$) normalized with the volume enclosed by the surface. Formally, its definition is:

$$\mathrm{div}(\mathbf F) = \lim_{\mu(\Omega)\to0}\frac1{\mu(\Omega)}\int_{\partial\Omega}\mathbf F\cdot\mathbf{\hat n}\,dS$$

Now, this might look a little scary at a first glance, but it's just the formalization of what I just said. We have a volume, denoted by $\Omega$. The function $\mu(\cdot)$ denotes the measure of such volume, in the most intuitive way. The integral is just the outgoing flux of the vector field over the surface ($\mathbf{\hat n}$ is in fact the outward pointing normal to $\partial\Omega$, that is, the boundary of $\Omega$).

So we divide this integral by the measure of $\Omega$. As $\Omega$ gets smaller, approaching a point ($\mu(\Omega)\to0$), we get the divergence (both the integral and $\mu(\Omega)$ approach $0$). We divide by $\mu(\Omega)$ because otherwise we would get the "absolute" value of the outgoing flux, which is just $0$ when dealing with a point. What we're interested in is the outgoing flux relatively to the volume, if that makes sense to you.

Okay, maybe I got a little bit technical and pretty much lost the "layman's terms" part. Let me go back to the intuitive explanation: in short, the divergence is just a measure of the "outgoingness" of the vector field. This has many, many applications in physics, way beyond the most intuitive one in fluid dynamics I provided before. For instance, one of Maxwell's equations is $\mathrm{div}(\mathbf E)=\frac\rho{\varepsilon_0}$, where $\mathbf E$ is the electric field, $\rho$ is the charge density and $\varepsilon_0$ is the vacuum permittivity.

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Okay, but what about the curl? It is a bit more tricky to understand, since it is a vector. Thus, to begin with, let us consider the $2$-dimensional case, where we take into consideration the curl of vectors lying on a plane. In this case, the curl is always orthogonal to the plane and thus can be treated as a scalar ($\mathrm{curl}(\mathbf F)\cdot\mathbf{\hat n}$, where $\mathbf{\hat n}$ is the unit vector normal to the plane).

The concept is analogous to that of the divergence. Here, the key concept is that of the circulation of the vector field along a closed curve on the plane, instead of the flux across a closed surface.

The scalar quantity $\mathrm{curl}(\mathbf F)\cdot\mathbf{\hat n}$ at a certain point $P$ represents the circulation along a closed path $\Gamma$ lying on the plane (and approaching $P$) normalized with the measure of the surface enclosed by $\Gamma$ (call it $A$). Formally, its definition is:

$$\mathrm{curl}(\mathbf F)\cdot\mathbf{\hat n} = \lim_{\mu(A)\to0}\frac1{\mu(A)}\oint_\Gamma\mathbf F\cdot\mathbf{\hat t}\,d\ell$$

Where of course $\mathbf{\hat t}$ denotes the tangent vector.

If the divergence measures how much a fluid is "diverging" (i.e. "getting away") from a certain point, the curl measures how much a fluid is "rotating" about a certain point. As an example, imagine a vortex. A property that we attribute to vortices is vorticity, that is intuitively how "rotating" the air is at a certain point. This is obviously computed using the curl.

Make sure you understand this $2$-dimensional definition before continuing to read. The extension to three dimensions is actually very simple: the vector curl is defined by its components along arbitrary direction $\mathbf{\hat n}$. The definition is basically the same. The fact is, $\mathbf{\hat n}$ is still the unit vector normal to the plane where $\Gamma$ lies, but in three dimensions virtually the vector field has components on any plane about a point.

The substancial difference is that now the curl does not always have the same direction, in general. This means that the elements of the fluid "rotate" around axes which are different at different points in the space.

To conclude, consider the simple example of the rotation of a rigid body. Let's take the Earth. It rotates about its rotational axis counterclockwise. If we were to calculate the curl of the velocity field here, the line that the curl vector lies on (i.e. its direction) would be the rotational axis, while the direction of rotation would be determined by the "sense" of such vector.

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This is the simplest way I've been able to put it, hope it helps. Unfortunately, it's very hard to talk about such concepts in a completely intuitive way, a little bit of formulas is always necessary.