4-3

If writing process is bothering you,

it's ok to write only correct answer in each question.

4. Evaluate / cy' dx + xy dy, where C consists of the arc of the parabola y = 27 from (0, 0) to (1, 1) and the line segments from (1, 1) to (0, 1) and from (0, 1) to (0, 0). Preview will appear here... Enter math expression here 5. Use Green's Theorem to find a formula for the area of the ellipse - + = 1 in terms of a and b. 2 2 O abn 6. A real-valued function f (x, y) is called harmonic if its second partial derivatives satisfy the equation = 0. True or false? If f is harmonic on a simply-connected region containing C, then | 97 of dr - of dy = 0. O True False 7. In which of the following cases can Green's Theorem be applied to a function f? Select all that apply. The domain of f is in IR (one-dimensional). The domain of f is in R (two-dimensional). The domain of f is in IR3 (three-dimensional). 8. True or false? The Fundamental Theorem for Line Integrals is a special case of Green's Theorem. O True False1. Describe the orientation of the curve sketched below, indicated by the arrows. 2 Positive orientation O Negative orientation 2. Determine the orientation of the ellipse given by 7 (t) = (3 sin(-2t), 5 cos(21)), 0 s t s *. The ellipse is positively oriented. The ellipse is negatively oriented. 3. Let F(x, y) = (y - cosy, r siny). Use Green's Theorem to compute / F . dr, where C is the circle (x - 2)2 + (y + 3)? = 9 traversed clockwise. [Hint: note that Green's Theorem requires the curve to be positively oriented. To apply Green's Theorem, consider the same circle traversed counterclockwise.] Preview will appear here... Enter math expression here9. Evaluate y da - I dy, where C'is the boundary of the rectangle (-5, -3 x 4, 7, oriented in the counterclockwise direction. Preview will appear here... Enter math expression here 10. True or false? Green's Theorem can be used to prove that if F = P : + Q j where P and Q have continuous ap aQ partial derivatives and ay throughout an open simply-connected region D, then " is conservative on D. True O False