91Ó°ÊÓ

Line integrals are an essential concept in vector calculus that help in understanding how a vector field affects a curve. When performing a line integral, we integrate along a path or curve, which is denoted by the symbol \( \oint \) when the path forms a closed loop. Line integrals are used to calculate the work done by a force field as an object moves along a path or to find circulation and flux.

For example, in the given exercise, the line integral \( \oint (3y \, dx + 2x \, dy) \) represents integrating the expression over the closed curve \( C \), which is the boundary of the region \( 0 \leq x \leq \pi, 0 \leq y \leq \sin x \). This integral measures the work done by the vector field \( \mathbf{F} = (3y, 2x) \) along curve \( C \).

• The function \( M = 3y \) multiplies the differential \( dx \), indicating changes in the \( x \)-direction.
• The function \( N = 2x \) multiplies the differential \( dy \), indicating changes in the \( y \)-direction.

Such an integral can often be simplified using Green's Theorem, allowing us to convert it into a double integral, making it easier to evaluate.

Double integrals allow for the calculation of volume under a surface over a region \( R \). These integrals extend the concept of calculating area under a curve to a two-dimensional region. In the context of Green's Theorem, double integrals help in determining the area or the total contribution of a field over a defined region.

In the given exercise, after applying Green's Theorem, the line integral is converted into a double integral \( \int\int_R (2 - 3) \, dA = \int\int_R (-1) \, dA \). Here, the partial derivatives computed from the vector field give us a simplified version for calculating the integral by integrating over region \( R \), which is under the sine curve between \( x = 0 \) and \( x = \pi \).

• The negative sign in \( \int\int_R (-1) \, dA \) signifies subtraction, indicating the downward contribution or the negative orientation of the area.
• The region \( R \) is set up using the limits of the sine function within the specified range.

Through evaluating this double integral, we calculate the area under the sine curve precisely.

Partial derivatives are derivatives of multivariable functions with respect to one variable, holding other variables constant. They are crucial in understanding how a function changes at a point concerning each variable individually.

In Green's Theorem, partial derivatives are used to transform a line integral into a double integral. For the given vector field \( \mathbf{F} = (3y, 2x) \), we compute the partial derivatives with respect to \( x \) and \( y \):

• \( \frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(2x) = 2 \), showing how \( N = 2x \) changes in the \( x \)-direction.
• \( \frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(3y) = 3 \), showing how \( M = 3y \) changes in the \( y \)-direction.

These derivatives are used to compute \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 2 - 3 = -1 \), simplifying the line integral into an easier-to-evaluate double integral. Partial derivatives thus provide insight into the internal structure of the vector field and guide us through the simplification process.