Problem 32 - Thomas Calculus

Question

Evaluate the integral $$\oint_{C}-y^{3} d y+x^{3} d x$$ for any closed path \(C\)

Step-by-Step Solution

Answer

The integral evaluates to zero by Green's Theorem.

Step 1: Recognize the integral as a line integral

The integral is a line integral of the form \( \oint_{C} M \, dx + N \, dy \) where \( M = x^3 \) and \( N = -y^3 \). The path \( C \) is a closed curve.

Step 2: Check if the vector field is conservative

A vector field \( \mathbf{F} = (M, N) = (x^3, -y^3) \) is conservative if \( \frac{\partial N}{\partial x} = \frac{\partial M}{\partial y} \). Here, \( \frac{\partial N}{\partial x} = 0 \) and \( \frac{\partial M}{\partial y} = 0 \), which are equal. Hence, the field is conservative.

Step 3: Apply Green's Theorem

Green's Theorem relates the line integral around a simple closed curve \( C \) to a double integral over the plane region \( D \) bounded by \( C \). It states \( \oint_{C} M \, dx + N \, dy = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA \). We have previously found both partial derivatives to be 0.

Step 4: Evaluate the double integral

Since both partial derivatives from Green's Theorem are zero, the double integral becomes \( \iint_{D} 0 \, dA = 0 \), regardless of the path \( C \).

Step 5: Conclude the result

Therefore, the line integral around any closed path \( C \) is zero.

Key Concepts

Green's Theorem

Green's Theorem provides a powerful connection between a line integral around a closed path and a double integral over the region it encloses. This theorem states that, for a continuously differentiable vector field \( \mathbf{F} = (M, N) \), the line integral around a simple closed curve \( C \) is equal to the double integral of the difference in partial derivatives over the region \( D \) inside \( C \). Namely, \[ \oint_{C} M \, dx + N \, dy = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA \] Key applications of Green's Theorem:

• Transforming line integrals into area integrals, which are often easier to calculate.
• Confirming the circulation and flux of vector fields over a region.

In the provided exercise, using Green's Theorem simplified the problem significantly. By confirming that the partial derivatives were zero, the theorem allowed the conclusion that the integral evaluates to zero for any closed path \( C \). This reduction to zero showcases the elegance of employing Green's Theorem in appropriate scenarios.

Conservative Vector Fields

A vector field is considered conservative if it is the gradient of some scalar field. This means that for a vector field \( \mathbf{F} = (M, N) \), there exists a potential function \( f \) such that \( \mathbf{F} = abla f \). To check whether a given vector field is conservative, we use specific conditions:

• The curl of the vector field must be zero in two dimensions. This translates mathematically to \( \frac{\partial N}{\partial x} = \frac{\partial M}{\partial y} \).
• The domain of the vector field must be simply connected, meaning it has no holes.

In the original problem, the vector field \( \mathbf{F} = (x^3, -y^3) \) was confirmed as conservative because both \( \frac{\partial N}{\partial x} = 0 \) and \( \frac{\partial M}{\partial y} = 0 \). This zero value of the partial derivatives indicates that the field is conservative in the region of interest. A conservative vector field means any line integral evaluated over a closed path returns to its starting point with zero net value, consistent with the result showing the integral around \( C \) is zero.

Closed Paths

A closed path in calculus is a path where the start and end points are the same. When dealing with line integrals, the closed path plays a crucial role, especially if the vector field is conservative. For a conservative vector field:

• The line integral over any closed path is independent of the path and depends only on the endpoints. Since the endpoints coincide in a closed path, the integral evaluates to zero.

However, if the vector field is not conservative, the line integral over a closed path could result in different values, depending on the path taken. Green's Theorem simplifies such analyses in two-dimensional planes, as it replaces the challenging aspect of evaluating line integrals with easier double integrals. In the scenario discussed, the line integral was around a closed path \( C \), and since the vector field was conservative, the integral across \( C \) simplified to zero. Understanding the nature of closed paths helps evaluate such integrals more efficiently.