Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.
-$\mathbf { F } = \left( x - e ^ { \mathrm { x } } \cos y \right) \mathbf { i } + \left( x + \mathrm { e } ^ { \mathrm { x } } \sin \mathrm { y } \right) \mathbf { j } ; \mathrm { C }$ is the lobe of the lemniscate $\mathrm { r } ^ { 2 } = \sin 2 \theta$ that lies in the first quadrant.
A) 0
B) $\frac { 1 } { 2 }$
C) 1
D) $\frac { 1 } { 4 }$

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The Answer of Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C.
-$\mathbf { F } = \left( x - e ^ { \mathrm { x } } \cos y \right) \mathbf { i } + \left( x + \mathrm { e } ^ { \mathrm { x } } \sin \mathrm { y } \right) \mathbf { j } ; \mathrm { C }$ is the lobe of the lemniscate $\mathrm { r } ^ { 2 } = \sin 2 \theta$ that lies in the first quadrant.
A) 0
B) $\frac { 1 } { 2 }$
C) 1
D) $\frac { 1 } { 4 }$

Using Green's Theorem, Compute the Counterclockwise Circulation of F Around