Field Operators: Grad, Div, and Curl

Curl Operator (vector → vector)

Curl Definition

The curl measures the rotation (or curl) of a 3D vector field, and the result is another 3D vector field.

Given the general equation for a 3D vector field,

F(x,y,z) = M(x,y,z)î + N(x,y,z)ĵ + P(x,y,z)k̂      (3D Vector Field i.e. in R3)

where:

• x, y, and z are the coordinates of a point in space.
• î, ĵ , k̂ are the standard unit vectors along the x, y, and z axes.
• M, N, and P are scalar-valued functions that describe the components of the vector at each point.

The curl of F(x,y,z) is defined as the cross product of the del (nabla) operator and the vector field.

$\text{curl}F(x,y,z)=\nabla \times F$

= î($\partial$P/$\partial$y – $\partial$N/$\partial$z)

+ ĵ($\partial$M/$\partial$z – $\partial$P/$\partial$x)

+ k̂($\partial$N/$\partial$x – $\partial$M/$\partial$y)

note: Technically, The del operator is NOT A VECTOR, but by treating it as one we can use the cross product methodology to compute the curl. 

Picture: Curl Operator

Example 1: Computing the Curl

Let F(x,y,z) = M(x,y,z)î + N(x,y,z)ĵ + P(x,y,z)k̂ = (xy)î + (-sin z)ĵ + (1)k̂

$\text{curl}F(x,y,z)=\nabla \times F$

= î($\partial$P/$\partial$y – $\partial$N/$\partial$z)

+ ĵ($\partial$M/$\partial$z – $\partial$P/$\partial$x)

+ k̂($\partial$N/$\partial$x – $\partial$M/$\partial$y)

= î($\partial$1/$\partial$y – $\partial$(-sin z)/$\partial$z))

+ ĵ($\partial$(xy)/$\partial$z – $\partial$/$\partial$x))

+ k̂($\partial$(-sin z)/$\partial$x – $\partial$(xy)/$\partial$y))

= î(cos z) + ĵ(0) – k̂(x)

So, we took the curl of a vector field, and our result is another vector field: the curl vector field. 

We said the curl is an indicator of rotation but what does this look like on a graph?

Let’s look at Example 2 to understand this.

Example 2:

Let’s take the curl of a vector field in which the z component is zero.

Let F(x,y,z) = M(x,y,z)î + N(x,y,z)ĵ + (0)k̂ = 

the curl F(x,y,z) = ∇×F  

= î($\partial$P/$\partial$y – $\partial$N/$\partial$z)

+ ĵ($\partial$M/$\partial$z – $\partial$P/$\partial$x)

+ k̂($\partial$N/$\partial$x – $\partial$M/$\partial$y)

In this case P is zero and all the partials with respect to z are zero so

curl F(x,y,z) = ∇×F  

= k̂($\partial$N/$\partial$x – $\partial$M/$\partial$y)

= a curl vector field that is only pointing in the z direction, which is therefore perpendicular to the xy plane.

Example 3:

Let’s repeat example two on the specific vector field function: F(x,y,z) = <-y,x,0>

This is the 3D analogue of the rotating 2D vector field F(x,y) = <-y,x> which is shown in the graph below.

Picture: F(x,y) = <-y,x> Vector Field

Lets convert this 2D function graph into its analogous 3D function graph:

• So we want to  convert the graph of F(x,y) = <-y,x> into
• the graph for F(x,y,z) = <-y,x,0>.
• Imagine that you take an infinite number of vector fields like the one above and stack them up in the xy plane direction in 3D space.

Then, you get the graph below.

Picture: Graph of  F(x,y,z) = <-y,x,0>

So, you get it, you get a graph of an infinite stack of vector fields in the XY plane.

Now lets take the curl of F(x,y,z):

= ∇×F 

=  ∇×<-y,x,0>

= k̂($\partial$N/$\partial$x – $\partial$M/$\partial$y) 

= (1 + 1)k̂ = 2k̂

So the result is a Curl vector field with vectors of magnitude 2 pointing in the z direction.

Let’s incorporate these Curl vector fields onto our 3D vector field graph.

Picture: Curl of Vector Field

Voila.

Description of Graph Above

• An infinite number of xy planes stacked up the z axis.
• Each plane contains the same flow vector fields, embedded in the xy plane.
• Each plane has an associated curl vector field.
• The curl vector field has vectors emanating from the xy plane and pointing vertically upwards
• with a magnitude of 2 (identically for each plane).
• Using the Right Hand Rule and given the rotation direction (counterclockwise),
  • the curl vector will point up,
  • perpendicular to the rotating xy plane.
• The curl is a positive, another indication that the rotation is counterclockwise.

It Can Get Very Complicated

We simplified the 3D function graph above by providing a simple vector field with a zero z (k) component.

You can imagine how these can get a lot more complicated with no easy visual to help us make sense of the vector fields and their curl fields.

You really have to think about each singular curl vector resulting from each singular location in 3D space.

See the picture below.

Picture – Fluid Curl Vectors

Imagine all the points in 3D space being represented as infinitesimally small spheres.

• Each sphere is fixed at its center but can otherwise rotate in any direction.
• Each sphere will eventually line up along its axis of rotation.
• This will be perpendicular to the plane of rotation.
• The Right Hand Rule will dictate which way the vector points.
• The Vector lengths can change indicating differing rates of rotation.

The 2D Analog of a Curl

Let’s look at example 2 again:

Let F(x,y,z) = M(x,y,z)î + N(x,y,z)ĵ + (0)k̂.

$\text{The curl}F(x,y,z)$

= ∇×F  

= î($\partial$P/$\partial$y – $\partial$N/$\partial$z)

+ ĵ($\partial$M/$\partial$z – $\partial$P/$\partial$x)

+ k̂($\partial$N/$\partial$x – $\partial$M/$\partial$y)

P is zero and all the partials with respect to z are zero so

curl F(x,y,z) = ∇×F  

= k̂($\partial$N/$\partial$x – $\partial$M/$\partial$y)

We can define a scalar curl or 2D curl as

curl F(x,y)=∂N/∂x−∂M/∂y

This scalar value

• represents the magnitude of the rotation and
• it’s sign indicates the direction of the rotation
  • Positive if counterclockwise and
  • Negative if clockwise
• The sign also indicates the direction of the axis of rotation
  • Pointing up if counterclockwise and
  • Pointing down if clockwise

• This scalar value is essentially the $z$-component of the 3D curl vector, which is why it can fully describe the rotation for a 2D field.

For a 2D vector field the curl will be a scalar vector field.

This can be shown on an xy graph as a colored or contoured overlay.

See the great example shown in the graph below.

Graph: Example of 2D Vector Field with Colored Scalar Curl Field Overlay

Properties of the Curl

The Curl of the Gradient of a scalar function f(x,y,z) is a  <0,0,0> vector field

• ∇×∇f= <0,0,0> 

The Divergence of the Curl of a vector field F is always zero

• $\nabla$•($\nabla \times F)=\; 0$

The vector identity for the Curl of a Curl of a vector field F is:

• ∇×(∇×F) = ∇(∇⋅F)−∇²F  where
• ∇(∇⋅F) is the gradient of the divergence of A.
  • The divergence (∇⋅A) is a scalar function,
  • and the gradient (∇) of that scalar function produces a vector field.
  • This term represents the “source” or “flow” part of the vector field.
• ∇²F is the vector Laplacian of F.
  • It’s the Laplacian operator applied to each component of the vector field,
  • resulting in another vector field.
  • It represents the “diffusion” or “spread” part of the vector field.
• The derivation of this is quite complicated so I am not including it in this post.
• This identify is used to derive the wave equation from Maxwell’s electromagnetic equations.  

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