Problem 31 - Advanced Engineering Mathematics

Question

Let \(P\) and \(Q\) be continuous and have continuous first partial derivatives in a simply connected region of the \(x y\) -plane. If \(\int_{A}^{B} P d x+Q d y\) is independent of the path, show that \&t \(P d x+Q d y=0\) on every piecewise- smooth simple closed curve \(C\) in the region

Step-by-Step Solution

Answer

The integral \(\int_{C} P \, dx + Q \, dy = 0\) because the vector field is conservative.

Step 1: Understand Path Independence

Given that the line integral \( \int_{A}^{B} P \, dx + Q \, dy \) is independent of the path, it suggests an important property related to vector fields. In simpler terms, if two functions \( P \) and \( Q \) together describe a vector field, the work done moving from point \( A \) to \( B \) does not depend on the path taken between these points.

Step 2: Apply Green's Theorem

Green's Theorem connects a line integral around a simple closed curve \( C \) to a double integral over the region \( R \) it encloses: \( \int_{C} P \, dx + Q \, dy = \int \int_{R} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \). We want to show this equals zero for any closed curve \( C \).

Step 3: Evaluate the Condition for Path Independence

To be path-independent and, therefore, a conservative vector field, the expression \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \) must be zero everywhere in the region. This means that there are no sources or sinks inside \( C \) and the vector field is the gradient of some scalar field.

Step 4: Conclusion from Green's Theorem

Since \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 \) across the region, applying Green's Theorem, we find: \( \int \int_{R} 0 \, dA = 0 \). Therefore, \( \int_{C} P \, dx + Q \, dy = 0 \) for any piecewise-smooth simple closed curve \( C \). This confirms that the vector field does not accumulate any net work around a closed path.

Key Concepts

Green's Theorem

Green's Theorem is a fundamental concept in vector calculus that establishes a remarkable connection between a line integral around a closed curve and a double integral over the region it encloses. If you imagine walking around a park's pathway, Green's Theorem allows you to relate your journey on the path to the entire area of the park.

The theorem states:

• For a vector field defined by functions \( P \) and \( Q \), the line integral over a simple closed curve \( C \) is mathematically expressed as \( \int_{C} P \, dx + Q \, dy \).
• According to Green's Theorem, this line integral can also be expressed as a double integral over the region \( R \) enclosed by \( C \): \( \int \int_{R} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \).

For the theorem to apply:

• \( P \) and \( Q \) must have continuous derivatives in \( R \).
• The region \( R \) needs to be "simply connected," meaning it shouldn't contain any holes.
• The curve \( C \) should be oriented so that the region \( R \) is always to your left as you traverse \( C \).

This theorem simplifies the calculation of line integrals and reveals intrinsic properties of the vector field defined by \( P \) and \( Q \).

Path Independence

Path independence in the context of vector fields is an elegant property indicating that the integral from a point \( A \) to a point \( B \) is independent of the actual path taken. If the work done, or the integral \( \int_{A}^{B} P \, dx + Q \, dy \), doesn't change regardless of the route you choose, this path independence has significant implications.

For a vector field to be path-independent:

• The field is said to be "conservative." This means the vector field is the gradient of some scalar field, indicating no net work is done moving along any closed path.
• The condition for path independence is directly linked to the derivatives \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 \). If this condition holds, then the line integral is path-independent in the region.
• This is often seen in physical contexts like gravitational fields where the path taken between two points doesn't alter the work done by the field.

Understanding path independence is crucial not only in theoretical mathematics but also in applications involving energy conservation.

Conservative Vector Fields

Conservative vector fields embody an important class of vector fields where path independence holds true and are closely tied to mathematical concepts such as potential fields and scalar fields.

Key properties of conservative vector fields:

• A vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \) is conservative if it can be expressed as the gradient of some scalar function \( f \). This means \( \mathbf{F} = abla f \).
• The presence of a scalar potential function \( f \) implies that any line integral \( \int_{A}^{B} P \, dx + Q \, dy \) simplifies to \( f(B) - f(A) \). The path choice doesn't matter, only the endpoints do.
• In physics, conservative fields are often encountered as force fields like electrostatic or gravitational fields. These fields do not "lose" or "gain" energy in closed loops.
• For a vector field to be conservative in a simply connected region, it must satisfy \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 \), aligning with Green's Theorem.

Conservative vector fields simplify analyses in both theoretical and applied mathematics due to their predictable behavior across different paths.