0.2 Cartesian vectors and tensors: their calculus  (Page 7/9)

During the discussion of the divergence of a vector field, we showed that the above relation holds for an infinitesimal volume. Suppose now that a macroscopic volume of space is a composite of the infinitesimal regions. The total volume integral is the sum of the infinitesimal volume integrals. However, the contribution of $\mathbf{a}•\mathbf{n}dS$ from the touching faces of two adjacent elements of volume are equal in magnitude but opposite in sign since the outward normal points in opposite directions. Thus in a summation of $\mathbf{a}•\mathbf{n}dS$ , the only terms that survive are those on the outer surface S, i.e., the surface integral is over the exterior surfaces of the macroscopic region. Q.E.D.

If $\mathbf{a}=\nabla \varphi$ we have

where $\delta \varphi /\delta n$ denotes the derivative in the direction of the outward normal. If the scalar $\varphi$ is temperature, this equation says that at steady-state, the integral of the net sources of heat in the volume is equal to the flux across the external surfaces.

Stokes' theorem

On a surface let $C$ be the curve or a finite number of curves forming the complete boundary of an area $S$ . We assume that the surface is two-sided and that $S$ can be resolved into a finite number of regular elements. Choose a positive side of $S$ and let the positive direction along $C$ be that in which an observer on the positive side must move along the boundary if he is to have the area $S$ always on his left. At each regular point on the surface let $\mathbf{n}$ be the unit normal drawn toward the positive side. Let a and its first derivatives be continuous on $\mathbf{S}$ . Stokes' theorem states that the circulation around a closed curve is equal to the surface integral of the normal component of the curl.

We showed earlier the circulation around an infinitesimal, closed curve was equal to the normal component of the curl multiplied by the area of the enclosed surface. We will extend the earlier result for an infinitesimal closed curve enclosing an infinitesimal surface to a macroscopic curve and surface. The macroscopic surface will be subdivided into a composite of many infinitesimal regions where the earlier result apply. The summation of the normal component of the curl multiplied by the area of the element is equal to the surface integral of the normal component of the curl. However, the quantity $\mathbf{a}•\mathbf{t}ds$ from the touching sides of two adjacent surface elements have equal magnitude but opposite sign since the direction of the line integrals are in opposite directions. Thus in the summation of the circulations, the only terms that survive are the contribution of the external bounding curve, i.e., the circulation is around the exterior curve C bounding the surface S. Q.E.D.

The classification and representation of vector fields

We mentioned earlier that a solenoidal vector field is one where $\nabla •\mathbf{a}=0$ everywhere and an irrotational vector field is one where $\nabla \times \mathbf{a}=0$ everywhere. A vector field that is the gradient of a scalar $\mathbf{a}=\nabla \varphi$ is irrotational. If a vector field is both irrotational and solenoidal it is the gradient of a harmonic function, where $\nabla •\left(\nabla \varphi \right)={\nabla }^2\varphi =0$ . It can be proven that if a vector field is both irrotational and solenoidal, it is uniquely determined in a volume $V$ if it is specified over $S$ , the surface of $V$ . There other types of named vector fields are discussed by Aris.