Evaluate the line integrals by applying Green’s theorem. ∮C(-y d x+x d y), where C consists of l

step 1 All right, problem 151 using green serum, D -DX of D -Y of negative -Y, DA, and then simplifying

Evaluate the line integrals by applying Green’s theorem....

Evaluate the line integrals by applying Green’s theorem.

$\oint_{C}(-y d x+x d y),$ where $C$ consists of line segment $C_{1}$ from $(-1,0)$ to $(1,0),$ followed by the semicircular arc $C_{2}$ from $(1,0)$ back to $(1,0)$

Key Concepts

Line Integrals of Vector Fields

A line integral of a vector field involves integrating a function along a curve, combining the field’s components with differential elements along the path. In the context of vector fields, it measures quantities like work or circulation. The computation often requires either a direct parameterization of the curve or, when applicable, an application of a theorem like Green’s to convert the problem into a potentially simpler double integral over the enclosed region.

Closed Curves and Orientation

The concept of closed curves and their orientation is critical for applying Green’s Theorem. A closed curve defines the boundary of a region, and the orientation (typically counterclockwise) ensures that the interior is on the left as one traverses the curve. Correctly understanding and setting the orientation is essential because it affects the sign and validity of the resulting double integral when converting a line integral using Green’s Theorem.

Green's Theorem

Green’s Theorem is a fundamental result in vector calculus that relates the circulation of a vector field around a closed curve to a double integral over the region enclosed by the curve. It transforms a line integral into a double integral by equating the integral of the field’s components around the boundary with the double integral of a combination of their partial derivatives. This theorem is extremely useful when evaluating line integrals over piecewise smooth curves, especially when the geometry of the region simplifies the computation.

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