0.2 Cartesian vectors and tensors: their calculus  (Page 6/9)

If $\mathbf{A}$ is a tensor, the notation $div\mathbf{A}$ or $\nabla •\mathbf{A}$ is sometimes used for the vector $A_{ij,i}$ . The index notation is preferred for tensors.

The curl of a vector field

The symbolic vector product or cross product of the vector operator $\nabla$ and a vector field $\mathbf{a}\left(\mathbf{x}\right)$ is called the curl of the vector field. It is the vector

That this definition is a combination of the previously definitions for the $\nabla$ operator and the cross product can be seen be carrying out the operations.

The connection between the curl of a vector field and the rotation of the vector field (it is called rot $\mathbf{a}$ in some older texts) can be illustrated by calculating the circulation of the vector field around a closed curve. Consider a elementary rectangle in the 023 plane normal to 01 with one corner $P$ at ( $x_1$ , $x_2$ , $x_3$ ) and the diagonally opposite one $Q$ at ( $x_1$ , $x_2+d{x}_2$ , $x_3+d{x}_3$ ) as shown in Fig. 3.8. We wish to calculate the line integral or circulation around this elementary closed curve of $\mathbf{a}•\mathbf{t}ds$ , where $\mathbf{t}$ is the unit tangent to the curve. Now the line through $P$ parallel to 03 has tangent $-{\mathbf{e}}_{\left(3\right)}$ and the parallel side through $Q$ has tangent ${\mathbf{e}}_{\left(3\right)}$ , and each is of length $d{x}_3$ . Accordingly, they contribute to $\mathbf{a}•\mathbf{t}ds$ an amount

Similarly, from the other two sides, there is a contribution,

Thus writing $dA=d{x}_{2}d{x}_3$ , we have

and in the limit

The suffix 023 has been put on the integral sign to show that the line integral in a 023 plane, and the last equation shows that the circulation in the $O23$ plane is equal to the component of the curl in the $O1$ direction. This correspondence between the curl and circulation gives physical meaning the curl of a vector field. It is a measure of the circulation or rotation of the motion. There is a direction associated with circulation, rotation, and curl. If the circulation around a closed curve is in the direction of the closed fingers of the right hand, then the curl is in the direction of the thumb.

A vector field $\mathbf{a}$ for which $\nabla \times \mathbf{a}=0$ is called irrotational because the circulation about any closed curve vanishes.

The first of these states that if a vector field $\mathbf{b}$ is the curl of a vector, i.e., $\mathbf{b}=\nabla \times \mathbf{a}$ then the vector field $\mathbf{b}$ is solenoidal. The second states that if a vector field $\mathbf{b}$ is equal to the gradient of a scalar, i.e., $\mathbf{b}=\nabla \varphi$ then the vector field $\mathbf{b}$ is irrotational. The last identity has the Laplacian operator ${\nabla }^2=\nabla •\nabla$ operating on a vector. The result is a vector whose components are equal to the Laplacian of the components, if the coordinates are Cartesian. This may not be the case in curvilinear coordinates.

Green's theorem and some of its variants

The divergence theorem, also called the Gauss' theorem, or Green's theorem equates the volume integral of the divergence of a vector field $\mathbf{a}\left(\mathbf{x}\right)$ to the surface integral of the normal component of the vector.