91Ó°ÊÓ

A line integral is a type of integral where a function is evaluated along a curve. Imagine you're walking along a path, and at each point, you measure some quantity like temperature or elevation. The line integral essentially adds up all these measurements along the path. It's a powerful concept used in physics and engineering to assess quantities like work done by a force field on a particle moving through space.
A line integral can be expressed in the form \( \oint_{C} P\,dx + Q\,dy \) when working with vector fields, where \( P \) and \( Q \) are functions that represent the vector field components. In our case, \( P(x, y) = xy \) and \( Q(x, y) = x^2 \). This way, the line integral measures how this vector field interacts with the path \( C \).

• Path \( C \) is typically a closed curve.
• The line integral evaluates the entire field along this path.

A vector field assigns a vector to every point in a plane or space. It can represent various physical and mathematical quantities like velocity fields of a fluid or electromagnetic fields. For any point \( (x, y) \) in our vector field, there is a vector with components \( (P, Q) \).
In our exercise, the vector field is given by \( P(x, y) = xy \) and \( Q(x, y) = x^2 \).

• Each component \( P \) and \( Q \) describes a different aspect of the field at the point \( (x, y) \).
• Green's Theorem connects the line integral of the vector field over a closed path to a double integral over the plane region it encloses.

Polar coordinates are an alternative to Cartesian coordinates for representing points in a plane. Instead of using \( x \) and \( y \), polar coordinates use \( r \) (the radius) and \( \theta \) (the angle). They are particularly useful for circular or radial regions.
In our problem, we need to evaluate a double integral over a circular region in the right half-plane. Switching to polar coordinates simplifies this process.

• The circle \( x^2 + y^2 = 1 \) is easily parameterized with \( r = 0 \) to \( 1 \) and \( \theta = 0 \) to \( \pi \), covering only the right half.
• With polar coordinates, \( x \) becomes \( r\cos\theta \) and the area differential \( dA \) becomes \( r\, dr\, d\theta \).

A double integral is used to integrate over a two-dimensional area, and it's particularly handy when you need to compute the volume under a surface or to find accumulated quantities over a region.
To apply Green’s Theorem, we convert the line integral into a double integral over the region \( D \) enclosed by the curve \( C \). For our exercise, Green's Theorem simplifies the line integral to:\[iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA = \iint_D x \ dA \]

• In our particular case, the region \( D \) is circular, requiring a switch to polar coordinates.
• Evaluation proceeds by first integrating with respect to \( r \) and then \( \theta \).

The process ultimately evaluates to zero, which indicates the vector field has no net circulation around the closed path \( C \).