Curl and Divergence
GradientBefore we talk about curl and divergence, we have to define the gradient function. Let's be real: a gradient is just a fancy word for a derivative in vector form. You have a scalar function, you take its gradient, and you end up with a vector: a directional derivative telling you how the function is changing and in which direction.

How do you spot a gradient? It's usually represented with an upside down triangle followed by the scalar function. The gradient of f is ∇f.

How do you calculate a gradient? It's the sum of the partial derivatives with respect to each variable multiplied by the unit vector in the direction of that variable. Since a gradient is simply a vector created by partial derivatives in different directions, it retains many of the properties of the derivative.

Gradients are fun, but now that we know about them, we can do other cool things.CurlLet me give it to you straight: curl is the infinitesimal rotation of a a vector field. Now that we got the textbook definition out of the way... what does curl look like? Say you have a vector field and you place an infinitesimal particle in any given spot within the field. Chances are that particle is going to want to rotate at that infinitesimal moment; the curl of the vector field tells you what that circulation density is.

By definition, the curl is a vector, and it can only be applied to vectors. The direction of the curl is the axis of rotation (as determined by the right-hand rule), and its magnitude is the magnitude of the rotation.

How do you spot a curl? The curl of vector F is written as ∇×F.

How do you calculate a curl? The notation gives you clue. To get the curl vector, you find the cross product of the gradient operator with the vector function.DivergenceThe divergence is the curl's pretty cousin. It's easier to calculate and often more useful.

How do you spot a divergence? The divergence of vector F is written as ∇·F.

How do you calculate a divergence? The notation gives you clue here as well: just find the dot product of the gradient operator with the function.Laplace OperatorThe Laplacian is the operator given by the divergence of the gradient of a function.

How do you spot a Laplacian? The Laplacian of a function f is written ∇2f. To calculate it, take the gradient of the function first, then take the divergence of the result.Properties of Curl and DivergenceGiven that F is a scalar function, ∇×(∇F)=0.
Given that F is a vector function, ∇·(∇×F)=0.