タグ: normal line

SURFACE INTEGRALS

Fig. 6-3
Fig. 6-3

Let S be a two-sided surface having projection \cal R on the xy plane as in the adjoining Fig. 6-3. Assume that an equation for S is z = f(x, y), where f is single-valued and continuous for all x and y in \cal R. Divide \cal R into n subregions of area \Delta A_p,\ p = 1,\ 2,\ \dots,\ n, and erect a vertical column on each of these subregions to intersect S in an area \Delta S_p.

Let \phi (x, y, z) be single-valued and continuous at all points of S. Form the sum

\displaystyle \sum_{p=1}^{n}\phi(\xi_p, \eta_p, \zeta_p)\Delta S_p \cdots(29)

where (\xi_p, \eta_p, \zeta_p) is some point of \Delta S_p. If the limit of this sum as n \rightarrow \infty in such a way that each \Delta S_p \rightarrow 0 exists, the resulting limit is called the surface integral of \phi(x, y, z) over S and is designated by

\displaystyle \underset{S}{\iint}\phi(x, y, z)dS\cdots(30)

Since \Delta S_p = |\sec\gamma_p|\Delta A_p approximately, where \gamma_p is the angle between the normal line to S and the positive z axis, the limit of the sum (29) can be written

\displaystyle \underset{\cal R}{\iint}\phi(x, y, z)|\sec\gamma|dA\cdots(31)

The quantity |\sec\gamma| is given by

\displaystyle |\sec\gamma| = \frac{1}{|\bold{n}_p\cdot\bold{k}|} = \sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2}\cdots(32)

Then assuming that x = f(x, y) has continuous (or sectionally continuous) derivatives in \cal R, (31) can be written in rectangular form as

\displaystyle \underset{\cal R}{\iint}\phi(x, y, z)\sqrt{1 + \left( \frac{\partial z}{\partial x} \right)^2 + \left( \frac{\partial z}{\partial y} \right)^2}dxdy \cdots(33)

In case the equation for S is given as F(x, y, z) = 0, (33) can also be written

\displaystyle \underset{S}{\iint}\phi(x, y, z)\frac{\sqrt{(F_x)^2 + (F_y)^2 + (F_z)^2}}{|F_z|}dxdy\cdots(34)

The results (33) or (34) can be used to evaluate (30).

In the above we have assumed that S is such that any line parallel to the z axis intersects S in only one point. In case S is not of this type, we can usually subdivide S into surfaces S_1,\ S_2,\ \dots which are of this type. Then the surface integral over S is defined as the sum of the surface integrals over S_1,\ S_2,\ \dots

The results stated hold when S is projected on to a region \cal R of the xy plane. In some cases it is better to project S on to the yz or xz planes. For such cases (30) can be evaluated by appropriately modifying (33) and (34).