タグ: single-valued

STOKE’S THEOREM

Let S be an open, two-sided surface bounded by a closed non-intersecting curve C (simple closed curve). Consider a directed line normal to S as positive if it is on one side of S, and negative if it is on the other side of S. The choice of which side is positive is arbitrary but should be decided upon in advance. Call the direction or sense of C positive if an observer, walking on the boundary of S with his head pointing in the direction of the positive normal, has the surface on his left. Then if A_1,\ A_2,\ A_3 are single-valued, continuous, and have continuous first partial derivatives in a region of space including S, we have

\displaystyle \int_C[A_1dx + A_2dy + A_3dz] =\\\vspace{0.2in} \underset{S}{\iint}\left[ \left( \frac{\partial A_3}{\partial y} -\frac{\partial A_2}{\partial z} \right)\cos\alpha + \left( \frac{\partial A_1}{\partial z} -\frac{\partial A_3}{\partial x} \right)\cos\beta + \left( \frac{\partial A_2}{\partial x} -\frac{\partial A_1}{\partial y} \right)\cos\gamma \right]dS \cdots(38)

In vector form with \bold{A} = A_1\bold{i} + A_2\bold{j} + A_3\bold{k} and \bold{n} = \cos\alpha\bold{i} + \cos\beta\bold{j} + \cos\gamma\bold{k}, this is simply expressed as

\displaystyle \int_C \bold{A}\cdot d\bold{r} = \underset{S}{\iint}(\nabla\times\bold{A})\cdot\bold{n}dS\cdots(39)

In words this theorem, called Stoke’s theorem, states that the line integral of the tangential component of a vector \bold{A} taken around a simple closed curve C is equal to the surface integral of the normal component of the curl of A taken over any surface S having C as a boundary. Note that if, as a special case \nabla\times\bold{A} = 0 in (39), we obtain the result (28).

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).