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A line integral is essentially an extension of an ordinary integral. It involves integrating a function along a curve or path rather than along a straight line or over an interval. In the context of a line integral over a closed curve such as circle or the path enclosing a region, the integral can measure various physical quantities:

• Mass of wire if density varies along the wire
• Total work done by a force field in moving an object along a path
• Flux across a boundary of a domain

In Green’s Theorem, the line integral represents the work done by a vector field as you move around the boundary of a region. Here, we are given a line integral involving functions of two variables, such as \( M(x, y) = \tan^{-1} y \) and \( N(x, y) = -\frac{y^{2} x}{1+y^{2}} \).
Using the notation \( \oint_C M\,dx + N\,dy \), Green's Theorem helps us transform this line integral over the curve \( C \) into an easier-to-evaluate double integral over the region \( R \) enclosed by \( C \).

A double integral extends the concept of a single integral to functions of two variables, allowing for the calculation of volumes under a surface or area of a region. Unlike a single integral, which involves integrating a function of one variable along an interval, a double integral involves a two-dimensional domain.
In Green's Theorem, the line integral around a closed curve is related to a double integral over the region \( R \) enclosed by the curve. For the given problem, we set up a double integral in the form:\[\iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA.\]After substituting the partial derivatives, it boils down to a simpler form:\[\iint_R (-1) \, dA.\]This represents integrating the constant function \(-1\) over the area of the square, effectively calculating the area multiplied by \(-1\). For the region \( R \), which is a square with vertices at (0,0), (1,0), (1,1), and (0,1), the double integral resolves to \((-1) \cdot \text{Area}(R)\), equaling \(-1\).

Partial derivatives are used to understand how a function changes as each variable changes separately, holding others constant. They are crucial in multivariable calculus and integral to applying Green's Theorem effectively.

• **Partial Derivative of M**: In our problem, \( M(x, y) = \tan^{-1} y \), so finding \( \frac{\partial M}{\partial y} \) means looking at how \( \tan^{-1} y \) changes with \( y \): \[ \frac{\partial M}{\partial y} = \frac{1}{1+y^{2}}. \]
• **Partial Derivative of N**: For a function \( N(x, y) = -\frac{y^{2} x}{1+y^{2}} \), computing \( \frac{\partial N}{\partial x} \) involves examining how \( N \) changes with \( x \): \[ \frac{\partial N}{\partial x} = -\frac{y^{2}}{1+y^{2}}. \]

The difference of these derivatives, \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \), is then integrated over the region \( R \) as part of the double integral in Green’s Theorem. Understanding partial derivatives allows us to transform a potentially complex line integral into a more manageable form by using double integrals.