SKKU-Calculus-Sec-15-8 Stokes’ Theorem SGLee+이인행
15.8 Stokes’ Theorem by SGLee, 이인행
 |
* Stokes' Theorem
: a surface integral over an orientable surface S to a line integral over the boundary of S
15.8절에서는
Stokes' Theorem을 이용하여 선적분을 면적분으로, 면적분을 선적분으로 바꿀수 있는가를 묻는 문제
1.Evaluate 
using Stokes’ Theorem, given that {c} ^{} {} is the circle: 
that lies inside the cylinder 
and above the 
-plane.
http://matrix.skku.ac.kr/cal-lab/cal-15-8-1.html
Note that the curve of intersection is the circle at the plane 
.
Note : 위의 결과는 선적분한 결과이다.
http://sage.math.canterbury.ac.nz/home/pub/133/ : curl의 코드를 가져왔습니다.
Note : 위의 결과는 S의 면적분의 결과이다.
결과 값이 같음을 확인하여 Stokes' Theorem이 성립함을 알 수 있습니다.
2. Evaluate (a) directly (b) using Stoke's Theorem 
where C
is the ellipse
.
Let 
.
So 
and 
(a) Directly
(b) Using Stokes' Theorem
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3. Verify Stoke's Theorem for the vector field 
over an orientable surface 
which is the upper hemisphere 
and 
.
Note that the curve 
is 
, 
with positive orientation.
Since curl
, we have 
curl
.
Next we compute 
where 
, 
.
Hence 
.
Hence verified.
Note : 문제가 z>=0의 구간이 아니라 반구와 z=0인 면에서의 적분을 묻는 문제입니다.
4. Verify Stoke’s Theorem for a vector field 
over an orientable surface 
which
is the square 
in the 
.
[Hint] 
is a square of 
.
5. Verify Stoke’s Theorem for a vector field 
over a rectangle bounded by 
.
6. Evaluate 
, where 
and 
is the curve which is the intersection of 
and 
. (
is upward anticlockwise).
curl
and the curve 
is a boundary of 
on 
.
(Using Strokes' Theorem)
curl
.
이므로
7. We consider the vector field 
and the curve 
which is the boundary
of the triangle with vertices 
. Compute the work done by the force field 
in moving a particle along the curve 
. (First, the particle goes from 
to 
, and goes from 
to 
, finally goes from 
back to 
).
Consider the region 
. The 
is the boundary of 
. Its unit normal vector is 
. 
.
Using Stokes' Theorem, we have
The area of 
is 
.
8. Evaluate 
. Here 
and 
is a triangle with vertices 
.
curl 
.
Here we have 
.
,
,
curl
각각의 선분에 대해서 적분을 하면 매우 복잡한 과정이 되므로 Stokes' Theorem을 이용하여
주어진 선적분을 Domain 을 xy 평면으로하는 계산이 간편한 면적분으로 바꿀수 있습니다.
