Question

A vector field \(\vec{F}\) and a closed curve \(C,\) enclosing a region \(R,\) are given. Verify Green's Theorem by evaluating \(\oint_{C} \vec{F} \cdot d \vec{r}\) and \(\iint_{R}\) curl \(\vec{F} d A,\) showing they are equal. \(\vec{F}=\left\langle 0, x^{2}\right\rangle ; C\) the triangle with corners at (0,0),(2,0) and (1,1).

Short answer

Both integrals evaluate to \(\frac{2}{3}\), verifying Green's Theorem.

Step-by-step solution

Understand Green's Theorem

Green's Theorem relates a line integral around a simple closed curve \(C\) to a double integral over the plane region \(R\) bounded by \(C\). The theorem states \(\oint_C \vec{F} \cdot d\vec{r} = \iint_R ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} )\, dA\) for \(\vec{F} = \langle P, Q \rangle\).

Identify Components of \(\vec{F}\)

Given \(\vec{F} = \langle 0, x^2 \rangle\), identify \(P = 0\) and \(Q = x^2\).

Compute the Line Integral \(\oint_C \vec{F} \cdot d\vec{r}\)

The line integral \(\oint_C \vec{F} \cdot d\vec{r}\) is split into three segments, one for each side of the triangle. Compute the integral piecewise for segments (0,0) to (2,0), (2,0) to (1,1), and (1,1) to (0,0). For each line segment, find the parameterization of the path and evaluate the integral.

Parameterize and Evaluate Line Integrals

1. For (0,0) to (2,0): \( x=t, y=0 \) for \( t \) from 0 to 2. \( d\vec{r} = \langle 1, 0 \rangle \, dt\), so the integral is 0.2. For (2,0) to (1,1): Use \( x=2-t, y=t \) for \( t \) from 0 to 1. \( d\vec{r} = \langle -1, 1 \rangle \, dt \). Compute \(\int_0^1 (2-t)^2 \, dt\), which results in \( \frac{7}{3} \).3. For (1,1) to (0,0): \( x=t, y=t \) from 0 to 1. \( d\vec{r} = \langle 1, 1 \rangle \, dt \). Integrate \(\int_0^1 t^2 \, dt \), which results in \(- \frac{1}{3} \).4. Sum results: \(0 + \frac{7}{3} - \frac{1}{3} = \frac{2}{3} \).

Compute Curl of \(\vec{F}\) and Evaluate Double Integral

The curl of \(\vec{F}\), \(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\), is \(2x\). For the region \(R\), a triangle with vertices (0,0), (2,0), and (1,1), set up the double integral \(\iint_R 2x \, dA\). Use bounds: \(0 \leq x \leq 2 \) and \(0 \leq y \leq x/2 \) for lower section, and \(1 \leq x \leq 2 \), \(x-1 \leq y \leq 1 \) for upper section.

Evaluate the Double Integral

Compute the double integral: 1. Lower triangle: \(\int_0^1 \int_0^x 2x \, dy \, dx \) gives \(\frac{1}{3} \).2. Upper triangle: \(\int_1^2 \int_{x-1}^1 2x \, dy \, dx \) gives \(\frac{1}{3} \).Total of both is \(\frac{2}{3}\).

Conclusion from Green's Theorem

Compare results from Steps 4 and 6. The line integral \(\oint_C \vec{F} \cdot d\vec{r}\) is \(\frac{2}{3}\) and the double integral of the curl \(\iint_R \, 2x \, dA\) is also \(\frac{2}{3}\), verifying Green's Theorem.

Key concepts

Vector Field

A vector field is essentially a function that assigns a vector to each point in a particular space. In two dimensions, this can be visualized as an array of arrows, each possessing a specific direction and magnitude. Mathematically, we represent a vector field on the plane as \( \vec{F}(x, y) = \langle P(x, y), Q(x, y) \rangle \). Here, \( P \) and \( Q \) are functions of \( x \) and \( y \) which define the components of the vectors.

In our exercise, the given vector field is \( \vec{F}(x, y) = \langle 0, x^2 \rangle \). This shows that the field has no horizontal component (since \( P(x, y) = 0 \)) and a vertical component \( Q(x, y) = x^2 \). The magnitude and direction of these vectors change depending on the value of \( x \), illustrating how vector fields can model varying conditions across different regions.

Understanding vector fields is crucial for visualizing phenomena such as fluid flow and electromagnetic fields, offering insight into how forces and currents behave in varying spaces.

Line Integral

A line integral, in the context of vector fields, measures the accumulation of a field along a specific path or curve. It can be thought of as summing the effect of the vector field along the trajectory. Mathematically, we denote it as \( \oint_C \vec{F} \cdot d\vec{r} \), where \( C \) is the curve over which we integrate.

This integral evaluates how much the vector field "pushes" along the curve. Parametrize the curve to compute the line integral segment by segment. For example, in our exercise, we have a triangular path, and the line integral was evaluated over each side of the triangle.

Every segment requires understanding its path parameters, such as \( d\vec{r} \), the differential element along the path, and how \( \vec{F} \) interacts with it. The line integral gives us a single value which represents the cumulative interaction of the vector field along \( C \), vital in physics for determining work done by a field along a path.

Curl

The curl of a vector field is a measure of its tendency to induce rotation around a point. For a two-dimensional field described by \( \vec{F}(x, y) = \langle P(x, y), Q(x, y) \rangle \), the curl is a scalar given by \( abla \times \vec{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \).

Conceptually, if you imagine releasing a tiny paddlewheel into a fluid, the curl determines how strongly the wheel would spin. In our problem, \( P = 0 \) and \( Q = x^2 \), so the curl is \( 2x \). This shows a tendency for rotation that varies linearly with \( x \), implying regions of stronger or weaker spin.

Curl is fundamentally important in fluid mechanics and electromagnetism, offering insights into the behavior of vector fields in different regions. Evaluating the curl allows us to delve deeper into field properties such as circulation and vorticity, enhancing our understanding of complex flow fields.

Double Integral

A double integral, in the context of Green's Theorem, computes the cumulative effect of a field across an area. It is expressed as \( \iint_R ( abla \times \vec{F} ) \, dA \), where \( R \) is the region enclosed by the curve \( C \), and \( dA \) represents an infinitesimal area element.

Evaluating this requires setting appropriate bounds for \( x \) and \( y \), based on the geometry of the region \( R \). In the given exercise, \( R \) is a triangular region, and the integral focuses on calculating the net effect of the curl\( , 2x \) in this case, over all points in \( R \).

By integrating the curl over \( R \), we effectively relate local rotational effects to a global property, corresponding to the circulation around the boundary \( C \). The result of double integration is crucial in confirming Green's Theorem, illustrating the relationship between line integrals and double integrals over a plane region. This principle serves as a foundation for understanding broader concepts like divergence and Maxwell's equations in physics.