0.2 Cartesian vectors and tensors: their calculus  (Page 3/9)

If a surface is not closed, it normally has a space curve as its boundary, as for example a hemisphere with the equator as boundary. It has two sides if it is impossible to go from a point on one side to the other along a continuous curve that does not cross the boundary curve. The surface is sometimes called the cap of the space curve.

If $S$ is a piece-wise smooth surface with two sides in three-dimensional space, we can divide it up into a large number of small surface regions such that the dimensions of the regions go to zero as the number of regions go to infinity. If the regions fill the surface and are not overlapping, then sum of the areas of the regions is equal to the area of the surface. If the function, $F$ is defined on the surface, it can be evaluated for some point of each subregion of the surface and the sum $\Sigma F\Delta S$ computed. The limit as the number of regions go to infinity and the dimensions of the regions go to zero is surface integral of $F$ over $S$ .

The traditional symbol of the double integral is retained because if the surface is a plane or the surface is projected on to a plane, then Cartesian coordinates can be defined such that the surface integral is a double integral of the two coordinates in the plane. Also, two surface coordinates can define a surface and the double integration is over the surface integrals.

In transport phenomena the surface integral usually represents the flow or flux of a quantity across the surface and the function $F$ is the normal component of a vector or the contracted product of a tensor with the unit normal vector. Thus one needs to know the direction of the normal in addition to the differential area to calculate the surface integral. Consider the case of a surface defined as a function of two surface coordinates.

To see how we arrive at this result, recall the partial derivatives of the coordinates of a curve with respect to a parameter is a vector that is tangent to the curve. The magnitude is

The vector product has a magnitude equal to the product of the magnitudes and the sine of the angle between the vectors. This gives us the area of a parallelogram corresponding to the area of the differential region.

The two tangent vectors in the direction of the surface coordinates lie in the tangent plane of the surface. Thus the direction of the vector product is perpendicular to the surface. Inward or outward direction for the normal has not yet been specified and will be determined by the sign.

Volume integrals

The volume integral of a function $F$ over a volumetric region of space $V$ is the limit of the sum of the products of the volume of small volumetric subregions of $V$ and the function $F$ evaluated somewhere within each subregion.

Change of variables with multiple integrals

In Cartesian coordinates the elements of volume dV is simply the volume of a rectangular parallelepiped of sides $d{x}_1$ , $d{x}_2$ , $d{x}_3$ and so