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Chapter 6: Q30MP (page 338)

鈭�C鈥�(x2鈭�y)dx+(x+y3)dy: where Cis the parallelogram with vertices (0,0),(2,0),(1,1),(3,1).

Short Answer

Expert verified

The solution is鈭�cx2鈭�ydx+x+y3dy=4

Step by step solution

01

Given Information.

The given information is,鈭�Cx2鈭�ydx+x+y3dy

02

Definition of Green’s Theorem.

Green's theorem connects a line integral around a simple closed curve to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

03

Find the solution.

鈭�cx2鈭�ydx+x+y3dy

Use Green鈥檚 Theorem role="math" 鈭�A鈥�鈭�Q鈭�x鈭�鈭�P鈭�ydxdy=鈭�鈭�APdx+Qdy, where is the boundary of the area A.

It can be seen that

P=x2鈭�y鈫�鈭�P鈭�y

=-1

Q=x+y3鈫�鈭�Q鈭�x

=1

Since the contour is the parallelogram with the given vertices, the area Ais then the area of the paralleologram.

鈭�cx2鈭�ydx+x+y3dy=2(2)(1)

=4

鈭�V脳dr=5(1鈭�0)2+(0鈭�1)22

=10

Hence, the solution is 鈭�cx2鈭�ydx+x+y3dy=4.

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Most popular questions from this chapter

Given 蠒=x2-yz and the point (3,4,1) find

(a) 鈭�蠁 at P ;

(b) a unit vector normal to the surface 蠁=5 at P ;

(c) a vector in the direction of most rapid increase of 蠁 at P;

(d) the magnitude of the vector in (c);

(e) the derivative of at in a direction parallel to the liner=i-j+2k+(6i-j-4k)t.

Use Green鈥檚 theorem (Section 9) to do Problem 8.2.

鈭�V鈭�苍诲蟽over the entire surface of a cube in the first octant with edges of length 2along the coordinate axes, whereV=(x2鈭�y2)i+3yj鈭�2xzk.

鈭�(鈭�脳V).nd蟽over the surface consisting of the four slanting faces of a pyramid whose base is the square in the (x,y) plane with corners at (0,0),(0,2),(2,0),(2,2), and whose top vertex is at (1,1,2) whereV=(x2z-2)i+(x+y+z)j-xyzk.

In the figure u1is a unit vector in the direction of an incident ray of light, and u3and u2 are unit vectors in the directions of the reflected and refracted rays. If u is a unit vector normal to the surface AB, the laws of optics say that 胃1=胃3 and n1蝉颈苍胃1=n2蝉颈苍胃2 , where n1 and n2are constants(indices of refraction). Write these laws in vector form (using dot or cross products).
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