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. 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 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.