Sketch of solutions 2022 Calculus 2-2 B1 1. Use the change of variables to evaluate cos where is the region bounded by the ellipse . ⇔ if (변수 변환) sol) ⇒ cos cos ≤ cos ≤ cos sin sin 2. Find the center of mass of a wire that lies along the curve r j k , ≤ ≤ , if the density is . Sol) Let r t t t ≤ t ≤ ⇒ ∥ ′ ∥ ⇒ ∴ ⇒ ∴ And by symmetry (∵ is independent of ) i j k and be a curve given by Let F 3. r cos i sin j k , ≤ ≤ . Evaluate F∙ r . sol) : simply connected open region Let , , and . Then the following equalities hold in , , , i.e., ∇×F . Thus, F is conservative in . Let cos sin ≤ ≤ . Then and are curves from to inside . By the independence of line integral of in we have F∙ r F∙ r 4. Use the Green’s Theorem to evaluate the line integral of F tan i j along the polar curve cos , ≤ ≤ . Sol) Let ∪ where cos ≤ ≤ ⇒ ≤ ≤ ∪ F ∙ k sin cos sin F∙ cos ⇒ F ∙ F ∙ k F ∙ F ∙ ′ ∙ 5. Evaluate the line integral where consists of line segments from to , from to and from to . sol) Let be the line segment from to , the line segment from to , the line segment from to . Then ≤ ≤ => , ≤ ≤ ≤ ≤ => => , . Thus, . 6. Find the area of the part of the surface that lies inside the elliptic cylinder . sol) 이므로 ⟨ ⟩ , ≤ 곡면 S : ≤ 변수변환 ( ⇒ ≤ 7. Let be the surface obtained by revolving the curve ≥ , about the -axis. Evaluate the flux of F i j k across where is oriented upward. sol) r ⟨ cos cos cos sin sin ⟩ ≤ ≤ ≤ ≤ r ⟨ sin cos sin sin cos ⟩ r ⟨ cos sin cos cos ⟩ r × r ⟨ cos cos cos cos cos sin cos sin ⟩ F ∣ ∙ r × r cos sin The flux of F i j k across is F ∙ n F ∙ r × r cos sin cos sin cos sin sin . 8. Consider a polar curve cos ≤ ≤ , in the -plane. Let be the surface obtained by revolving the curve curl F ⋅ S about the -axis and let F i sin cosh j cos k . Evaluate where is oriented in the direction of the positive -axis. (문제 오류, oriented inward, that is, oriented in the direction to the origin.) sol) curl F ∙ i , 라 할 때 ′ ∣ ≤ 라 놓으면 ′ 따라서 스톡스 정리에 의해 F ⋅ r curl F ⋅ S curl F ⋅ , ′ ′ ′ 이므로 ′ (반지름 원판 ′의 넓이: ) i j k . Evaluate the flux Let F 9. F ⋅ S where is the part of the paraboloid ≤ ≤ oriented upward. ≤ ≤ , div F sol) ′ ≤ 인 곡면으로 막은 후 확장된 발산 정리 사용 ⇒ ′는 원점을 포함하는 닫힌 곡면이므로 F ⋅ S (교재 16.9절 ex3) 참고) ′ 1) F ⋅ S ∙ ′ 2) , ∙ ′ ′ ≤ 3) 1)과 2)에 의하여 F ⋅ S ∙ ′ 10. (a) 행렬 에 대해 det det 를 구하시오. (b) 크레머 공식 (Cramer’s Rule)을 이용하여, 다음 연립방정식의 해 중 의 값을 구하시오. Sol) (a) det ⇒ det det det det (b) ) det det det ∴ det
