Need help with part b). Please explain each step and neatly type up. Thank you :) 7. Suppose that C is the positively oriented square with vertices (-1, 1), (1, 1), (1, 1),and (-). LetF(x, y) = P(x, y)i + Q(x, y)jwhere(a)(b)P(x, y) = (cos(x) + x²)yx³3to a double integral.Q(x, y) = 1 + x +(*) Applying the Green's theorem, transform[.F.(*) Find the value ofF. dr[FF. dr

Answered: Suppose that C is the positively…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Need help with part b). Please explain each step and neatly type up. Thank you :)

7. Suppose that C is the positively oriented square with vertices (-1, 1), (1, 1), (1, 1),
and (-). Let
F(x, y) = P(x, y)i + Q(x, y)j
where
(a)
(b)
P(x, y) = (cos(x) + x²)y
x³
3
to a double integral.
Q(x, y) = 1 + x +
(*) Applying the Green's theorem, transform
[.F.
(*) Find the value of
F. dr
[F
F. dr

Expert Solution

Step 1

If $F = P (x, y) i + Q (x, y) j$ is a vector-valued function, then according to Green's theorem $\int_{C} F \cdot d r = \iint_{S} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d x d y$ , where C is the boundary of the surface. In this problem $P (x, y) = (\cos x + x^{2}) y$ , $Q = 1 + x + \frac{x^{3}}{3}$ and C is the positively oriented curve with vertices $(- \frac{π}{2}, - \frac{π}{2})$ , $(\frac{π}{2}, - \frac{π}{2})$ , $(\frac{π}{2}, \frac{π}{2})$ , $(- \frac{π}{2}, \frac{π}{2})$ . We have to find the value of $\int_{C} F \cdot d r$