Find the work done by the force field F on a particle moving...

Find the work done by the force field $F$ on a particle moving along the given path.
$\mathbf{F}(x, y)=-y \mathbf{i}-x \mathbf{j}$
C: counterclockwise along the semicircle $y=\sqrt{4-x^{2}}$ from (2,0) to (-2,0)
(FIGURE CAN'T COPY)

Key Concepts

Fundamental Theorem for Line Integrals

This theorem provides a method to evaluate line integrals of conservative vector fields by simply subtracting the values of the potential function at the terminal points of the path. It significantly simplifies calculations by bypassing the need for explicit parameterization and integration along the curve.

Potential Function

A potential function is a scalar function whose gradient yields the conservative vector field. In the context of work and energy, it allows for the simplification of calculations involving line integrals, since the work done over any curve in the field equals the difference in the potential function evaluated at the endpoints.

Line Integral of a Vector Field

A line integral calculates the cumulative effect of a vector field along a prescribed path. It is a fundamental concept in vector calculus for quantifying work done by a force field when an object travels along a curve, and it requires integrating the dot product of the field and the curve's differential element.

Conservative Vector Field

A vector field is classified as conservative if it can be expressed as the gradient of some scalar potential function. This property implies that line integrals over the field are path-independent, meaning the work done by the force field depends only on the endpoints of the path, rather than the specific route taken.

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