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Unformatted text preview: ation. Therefore, we can apply Green’s Theorem to the line integral: xy 2 dx + x3 dy =
C R (Qx − Py ) dA ∂3 ∂ (x ) − (xy 2 ) ∂x ∂y 3x2 − 2xy dA
3 0 (Green’s Theorem) dA =
D =
D 2 =
0 2 3x2 − 2xy dydx (D is a rectangle.) =
0 (9x2 − 9x) dx 2 = 6. Remark: The Fundamental Theorem says that the line integrals only depend on end points (the boundary of the domain). and the Green’s Theorem says the integration on the boundary can determine the the integration on the whole region. If you learn some knowledge on manifold, you will know the diﬀerential forms of Fundamental Theorem, Green’s Theorem are the same. Of course, this is not in the syllabus. 195 5. Evaluate
C (x5 − y 5 ) dx + (x5 + y 5 ) dy , where C is the boundary with positive orientation of the region between the circles x2 + y 2 = a2 and x2 + y 2 = b2 , where 0 < a < b. Solution Let D be the ring region enclosed by two concentric circle of radius a and b respectively. In polar coordinates, D : a ≤ r ≤ b, By Green’s Theorem, (x5 − y 5 ) dx + (x5 + y 5 ) dy
C 0 ≤ θ ≤ 2π . =
D ∂5 ∂5 (x + y 5 ) − (x − y 5 ) ∂x ∂y (5x4 + 5y 4 ) dA dA (Green’s Theorem) =
D =5
D 2π (x2 + y 2 )2 − 2x2 y 2 dA
b a (Completing the Square) =5
0 2π b (r 2 )2 − 2(r cos θ )2 (r sin θ )2 r drdθ r 5 1 − 2(cos θ )2 (sin θ )2 drdθ
2π 0 (Polar Coordinates) =5
0 b a a =5 = = = = r 5 dr 1 − 2(cos θ )2 (sin θ )2 dθ 1− 1 sin2 2θ dθ 2 31 + cos 4θ dθ 44 (Separate Variables) (sin 2θ = 2 sin θ cos θ ) (cos 4θ = 1 − 2 sin2 2θ ) 56 (b − a6 ) 6 56 (b − a6 ) 6 56 (b − a6 ) 6 5 π (b6 − a6 ) 4 2π 0 2π 0 3 π 2 2
8 D 6 4 2 ab
5 5 10 2 4 6 8 196 Practice Exercises I:
Here are some questions appeared in past years’ tutorials. I attach them here, just for reference: √ √ Pra 1.1 Compute the work done by the force ﬁeld F = x y i + 2y x j on a particle that moves from (1, 0) to (0, 1) counterclockwise along the C , where C consists of the shortest arc of the circle x2 + y 2 = 1 and then from (0, 1) to (4, 3) along a straight line. √ Ans: (32 3 + 66)/5 Pra 1.2 Let F(x, y ) = (ex sin y − y ) i +(ex cos y − x − 2) j and C be the curve given by r(t) = t i + t2 j, 0 ≤ t ≤ 1. Show that F is conservative, and hence calculate Ans: e sin(1) − 3 Pra 1.3 (i) Show that there does not exist a vector ﬁeld G on R3 such that curl G = xy 2 i + yz 2 j + zx2 k. (ii) Show that div F = 0 for any vector ﬁeld of the form F(x, y, z ) = f (y, z )i + g(x, z )j + h(x, y )k. Pra 1.4 Let F(x, y, z ) = 4xez i + cos y j + 2x2 ez k and C the curve with vector equation given by r(t) = t i + t2 j + t4 k, 0 ≤ t ≤ 1. Find a function f such that F = ∇f . Hence, or otherwise, evaluate
C C F • dr. F • dr. Ans: 2e + sin 1 Practice Exercises II:
1. Thomas’ Calculus Section 16.1  16.4. 2. Your Second Textbook: Advanced Engineering Mathematics. In the book, the author brieﬂy introduced the chapters from vector space to surface integral in 2 chapters (chapter 9 and 10). This is a good resource for you to review what you have learnt in the lecture, and contains many good exercises. Find out more useful resource from:
https://cid957f4598794dd719.skydrive.live.com/browse.aspx/Reference%20Books/Text%20Books 197...
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This note was uploaded on 05/05/2009 for the course MA MA1505 taught by Professor Ma1505 during the Spring '09 term at National University of Juridical Sciences.
 Spring '09
 MA1505

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