タグ: simple closed curve

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).

SIMPLE CLOSED CURVES. SIMPLY AND MULTIPLY-CONNECTED REGIONS

A simple closed curve is a curve which does not intersect itself anywhere. Mathematically, a curve in the xy plane is defined by the parametric equations x = \phi(t),\ y = \psi(t) where \phi and \psi are single-valued and continuous in an interval t_1 \le t \le t_2. If \phi(t_1) = \phi(t_2) and \psi(t_1) = \psi(t_2), the curve is said to be closed. If \phi(u) = \phi(v) and \psi(u) = \psi(v) only when u = v (except in the special case where u = t_1 and v = t_2), the curve is closed and does not intersect itself and so is a simple closed curve. We shall also assume, unless otherwise stated, that \phi and \psi are piecewise differentiable in t_1 \le t \le t_2.

If a plane region has the property that any closed curve in it can be continuously shrunk to a point without leaving the region, then the region is called simply-connected, otherwise it is called multiply-connected.

Fig6-xx

As the parameter t varies from t_1 to t_2, the plane curve is described in a certain sense or direction. For curves in the xy plane, we arbitrarily describe this direction as positive or negative according as a person traversing the curve in this direction with his head pointing in the positive z direction has the region enclosed by the curve always toward his left or right respectively. If we look down upon a simple closed curve in the xy plane, this amounts to saying that traversal of the curve in the counterclockwise direction is taken as positive while traversal in the clockwise direction is taken as negative.