Link to Current (updated) notes

Curl and Its Interpretation

The curl is the vector valued derivative of a vector function. As illustrated below, its operation can be geometrically interpreted as the rotation of a field about a point.

For a vector-valued function of $(x,y,z)$ :

$$\vec{v}(x,y,z) =\vec{v}(\vec{x}) = (v_1(\vec{x}) , v_2(\vec{x}) , v_3(\vec{x})) = v_1(x,y,z) \hat{i} + v_2(x,y,z) \hat{j} + v_3(x,y,z) \hat{k}$$ (13-3)

the curl derivative operation is another vector defined by:

$$\vec{v} = \nabla \times \vec{v} = \left( \left(\ensuremath{\frac{... ...}{\partial{x}}} - \ensuremath{\frac{\partial{v_1}}{\partial{y}}}\right) \right)$$ (13-4)

or with the memory-device:

$$\vec{v} = \nabla \times \vec{v} = \det \left( \begin{array}{ccc} ... ...nsuremath{\frac{\partial{}}{\partial{z}}}\ v_1 & v_2 & v_3 \end{array} \right)$$ (13-5)

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-13)

(html Lecture-13)

(xml+mathml Lecture-13)

Calculating the Curl of a Function Consider the vector function that is often used in Brakke's Surface Evolver program:

$$\vec{w} = \frac{z^n} {(x^2 + y^2)(x^2 + y^2 + z^2)^{\frac{n}{2}}} (y \hat{i} - x \hat{j})$$

This can be shown easily, using MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ , to have the property:

$$\nabla \times \vec{w} = \frac{n z^{n-1}} {(x^2 + y^2 + z^2)^{1 + \frac{n}{2}}} (x \hat{i} + y \hat{j} + z \hat{k})$$

which is spherically symmetric for $n=1$ and convenient for turning surface integrals over a portion of a sphere into a path-integral over a curve on a sphere.

1. Create vector function $\vec{w}$ above and visualize using the PlotVectorField3D function in MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ 's PlotField3D package.
2. The function will be singular for $n>1$ along the $z-axis$ , this singularity will be communicated during the numerical evaluations for visualization unless some care is applied.
3. Demonstrate the above assertion about $\vec{w}$ and its curl.
4. Visualize the curl: note that the field is points up with large magnitude near the vortex at the origin.
5. Demonstrate that the divergence of the curl of $\vec{w}$ vanishes for any $n$ .

One important result that has physical implications is that a the curl of a gradient is always zero: $f(\vec{x}) = f(x,y,z)$ :

$$\nabla \times (\ensuremath{\nabla}f) = 0$$ (13-6)

Therefore if some vector function $\vec{F}(x,y,z) = (F_x , F_y, F_z)$ can be derived from a scalar potential, $\ensuremath{\nabla}f = \vec{F}$ , then the curl of $\vec{F}$ must be zero. This is the property of an exact differential $df = (\nabla f) \cdot (dx, dy, dz) = \vec{F} \cdot (dx, dy, dz)$ . Maxwell's relations follow from equation 13-6:

$$\begin{split}0 = & \ensuremath{\frac{\partial{F_z}}{\partial{... ...math{\frac{\partial^2{f}}{\partial{x} \partial{y}}} \end{split}$$ (13-7)

Another interpretation is that gradient fields are curl free, irrotational, or conservative.

The notion of conservative means that, if a vector function can be derived as the gradient of a scalar potential, then integrals of the vector function over any path is zero for a closed curve--meaning that there is no change in ``state;'' energy is a common state function.

Here is a picture that helps visualize why the curl invokes names associated with spinning, rotation, etc.

Figure: Consider a small paddle wheel placed in a set of stream lines defined by a vector field of position. If the $v_y$ component is an increasing function of $x$ , this tends to make the paddle wheel want to spin (positive, counter-clockwise) about the $\hat{k}$ -axis. If the $v_x$ component is a decreasing function of $y$ , this tends to make the paddle wheel want to spin (positive, counter-clockwise) about the $\hat{k}$ -axis. The net impulse to spin around the $\hat{k}$ -axis is the sum of the two.
Note that this is independent of the reference frame because a constant velocity $\vec{v} =$   const. and the local acceleration $\vec{v} = \nabla f$ can subtracted because of Eq. 13-8.

Another important result is that divergence of any curl is also zero, for $\vec{v} (\vec{x}) = \vec{v} (x,y,z)$ :

$$\nabla \cdot (\nabla \times \vec{v}) = 0$$ (13-8)