SKKU-Calculus-Sec-15-3 Independence of the Path, SGLee+오교혁

15.3    Independence of the Path             by SGLee, HSKim, 오교혁

1-6. Determine whether or not the vector field is conservative. If it is conservative, find a potential of .

1.

Not conservative, since curl.

2.

Not conservative.

3.

Not conservative.

4.

Since, curl  is conservative.

―(1)

Taking partial derivative of  with respect to , we get    . Hence from (1) we get

―(2)，thus   ―(3)

Now taking partial derivative of  with respect to  we get . Hence from (3) we get

, a constant. Therefore, .

5.

Not conservative.

6.

Conservative, .

7. Show that the vector field  is not conservative.

Since curl is not conservative.

8. Determine whether the force field  is a conservative field.

* 위의 명령어로  표현돤 결과는 vector 로 표현되서 보기 편하다.

Ans: , hence  is not conservative field.

9. a. Prove that   is a conservative field.

b. Find its scalar potential .

c. Also find the work done in moving an object in this field from  to .

Scalar potential .

Work done.

10. Find the total work done in moving a particle by a force field

along the curve  form  to .

Plenary Speaker at AMC 2013, Bexco, Korea