タグ: partial derivative

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).