Problem 13 - Thomas Calculus

Question

Use Green's Theorem to find the counterclockwise circulation and outward flux for the field \(\mathbf{F}\) and curve \(C .\) \(\mathbf{F}=\left(y^{2}-x^{2}\right) \mathbf{i}+\left(x^{2}+y^{2}\right) \mathbf{j}\) \(C :\) The triangle bounded by \(y=0, x=3,\) and \(y=x\)

Step-by-Step Solution

Answer

Circulation is 9 and flux is -9.

Step 1: Understanding Green's Theorem

Green's Theorem relates a line integral around a simple closed curve \(C\) to a double integral over the plane region \(D\) bounded by \(C\). For circulation, the theorem is \[ \oint_{C} \mathbf{F} \cdot \mathbf{dr} = \iint_{D} \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA \]where \(\mathbf{F} = M\mathbf{i} + N\mathbf{j}\).

Step 2: Identify Components M and N

From the field \(\mathbf{F} = (y^2 - x^2)\mathbf{i} + (x^2 + y^2)\mathbf{j}\), identify \(M = y^2 - x^2\) and \(N = x^2 + y^2\).

Step 3: Compute Partial Derivatives

Calculate the partial derivatives: \(\frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2) = 2x\) and \(\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(y^2 - x^2) = 2y\).

Step 4: Setup Double Integral for Circulation

Substitute the partial derivatives into Green's Theorem for circulation:\[ \iint_D (2x - 2y) \, dA \]

Step 5: Determine the Limits of Integration

The region \(D\) is a triangle bounded by \(y=0\), \(x=3\), and \(y=x\). The limits can be set as: - For \(x\), from 0 to 3 - For \(y\), from 0 to \(x\)

Step 6: Evaluate the Double Integral for Circulation

Calculate \[ \int_{0}^{3} \int_{0}^{x} (2x - 2y) \, dy \, dx \]First, integrate with respect to \(y\): \[ \int_{0}^{x} (2x - 2y) \, dy = [2xy - y^2]_{0}^{x} = 2x^2 - x^2 = x^2 \] Next, integrate \(x^2\) with respect to \(x\): \[ \int_{0}^{3} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{3} = \frac{27}{3} = 9 \]

Step 7: Setup Double Integral for Flux

For the outward flux, Green's Theorem states:\[ \iint_D \left( \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} \right) \, dA \] Compute the partial derivatives that differ from the circulation case: - \(\frac{\partial M}{\partial x} = \frac{\partial}{\partial x}(y^2 - x^2) = -2x\) - \(\frac{\partial N}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^2) = 2y\)

Step 8: Evaluate the Double Integral for Flux

Set up and calculate \[ \iint_D (-2x + 2y) \, dA \]And solve using the same limits from Step 5:\[ \int_{0}^{3} \int_{0}^{x} (-2x + 2y) \, dy \, dx \]Integrate with respect to \(y\):\[ \int_{0}^{x} (-2x + 2y) \, dy = [-2xy + y^2]_{0}^{x} = -2x^2 + x^2 = -x^2 \]Then integrate \(-x^2\) with respect to \(x\):\[ \int_{0}^{3} -x^2 \, dx = \left[ -\frac{x^3}{3} \right]_{0}^{3} = -\frac{27}{3} = -9 \]

Step 9: Conclusion: Circulation and Flux Results

The result for the circulation using Green's Theorem is 9, and for the outward flux, it is -9. These results are computed from the double integrals over the triangular region given by the intersecting lines.

Key Concepts

Circulation

Circulation refers to the tendency of a vector field to "spin" around a closed curve. It measures how much a vector field rotates along a path enclosing an area. Green's Theorem provides a way to find circulation by converting a line integral over a curve into a double integral over the region it encloses.

In the given exercise, the circulation of the vector field \( \mathbf{F} = (y^2 - x^2)\mathbf{i} + (x^2 + y^2)\mathbf{j} \) around the triangle bounded by \( y=0 \), \( x=3 \), and \( y=x \) is calculated. This involves identifying the components \( M = y^2 - x^2 \) and \( N = x^2 + y^2 \), then calculating the difference of the partial derivatives \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \), yielding \( 2x - 2y \).

The circulation is then determined by integrating this expression over the triangular region using double integrals, resulting in a value of 9. It quantifies the net "twist" imparted by \( \mathbf{F} \) around the triangle.

Flux

Flux, in the context of vector fields and Green's Theorem, measures the quantity of a vector field "flowing" out of a closed curve. It is an important concept in vector calculus and is useful for understanding how fields move across areas.

For this exercise, the flux of the field \( \mathbf{F} \) across the boundary of the given region is calculated. Using Green's Theorem, the flux is represented by the double integral \( \iint_D \left( \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} \right) \, dA \). This involves computing \( \frac{\partial M}{\partial x} = -2x \) and \( \frac{\partial N}{\partial y} = 2y \), which when summed give \( -2x + 2y \).

Integrating this over the triangle provides the flux value of -9. This negative value indicates that more of the vector field flows into the triangular region rather than out, giving insight into the behavior of the vector field \( \mathbf{F} \).

Vector Fields

Vector fields are mathematical constructs that assign a vector to each point in space, describing the magnitude and direction at every location. These fields are often used to model physical quantities like velocity fields in fluids, gravitational fields, or electromagnetic fields.

In our exercise, the vector field \( \mathbf{F} = (y^2 - x^2)\mathbf{i} + (x^2 + y^2)\mathbf{j} \) is considered. This specific field involves a combination of \( i \) and \( j \) components, labeled as \( M \) and \( N \) respectively.

Analyzing vector fields involves identifying these components and understanding how they change across a region. Calculating their partial derivatives, as in this case, allows us to understand how they impact circulation and flux through Green's Theorem. The behavior of \( \mathbf{F} \) helps determine the rotational and outward tendencies on the boundary of the triangular region.

Double Integral

A double integral is a way of integrating over a two-dimensional area. It extends the concept of a single integral from one dimension to two, allowing for the calculation of quantities over an area, such as volume or total mass.

In the exercise, the double integral is a crucial tool for calculating both the circulation and the flux of the vector field \( \mathbf{F} \) over the triangular region. The process involves determining the limits of integration based on the boundaries of the triangle, defined by \( y=0 \), \( x=3 \), and \( y=x \).

The region is integrated over first with respect to \( y \) from 0 to \( x \), and then with respect to \( x \) from 0 to 3. This setup enables computing the expressions derived from the vector field's partial derivatives, offering solutions for both circulation and flux. The double integral simplifies the process of solving complex vector field problems using Green's Theorem.