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. \(\vec{F}=\langle x+y, 2 x\rangle ; C\) the curve that starts at \((0,1),\) follows the parabola \(y=(x-1)^{2}\) to \((3,4),\) then follows a line back to (0,1).

Short answer

The line and double integrals give equal values, verifying Green's Theorem.

Step-by-step solution

Compute Line Integral over Curve C

We start with the line integral \( \oint_{C} \vec{F} \cdot d\vec{r} \). This is divided into two segments. For the first segment, extract parametric equations for the parabola \( y=(x-1)^2 \) running from \((0,1)\) to \((3,4)\). Use \( x=t \), \( y=(t-1)^2 \) for \( t \in [0, 3]\). Compute \( d\vec{r} = \langle dt, d((t-1)^2) \rangle = \langle dt, 2(t-1) dt \rangle \). Evaluate the integral of \( \vec{F}\cdot d\vec{r} \) over this segment:\[\int_{0}^{3} \langle t+(t-1)^2, 2t \rangle \cdot \langle 1, 2(t-1) \rangle dt = \int_{0}^{3} (t+(t-1)^2) + 4t(t-1) dt\]Simplify and solve the integral, computing its result.

Compute Line Integral for the Second Segment

For the second portion of the curve (the straight line from \((3,4)\) to \((0,1)\)), use parametric equations \( x = 3 - 3s \), \( y = 4 - 3s \) where \( s \in [0, 1] \). Find \( d\vec{r} = \langle -3 ds, -3 ds \rangle \). Evaluate the integral of \( \vec{F} \cdot d\vec{r} \) over this segment:\[\int_{0}^{1} \langle (3 - 3s) + (4 - 3s), 2(3 - 3s) \rangle \cdot \langle -3, -3 \rangle ds\]Simplify and solve the integral, combining the results from both segments to find the total line integral.

Calculate the Curl of the Field

Compute the curl of the vector field \( \vec{F} = \langle x+y, 2x \rangle \). The curl in two dimensions is found using:\[\text{curl} \vec{F} = \frac{\partial (2x)}{\partial x} - \frac{\partial (x+y)}{\partial y}\]Evaluate this to find the scalar function that will be integrated over region \( R \).

Set Up Double Integral over Region R

Determine the limits of integration for the region \( R \), which is bounded by \( y = (x-1)^2 \) and the line \( y = -3x + 13 \). These give an intersection point \((3,4)\). The limits are from \( x = 0 \) to \( x = 3 \), and for each \( x \), \( y \) ranges from \((x-1)^2\) to \(-3x + 13\). Set up the double integral:\[\iint_{R} (\text{curl } \vec{F}) \, dA = \int_{0}^{3} \int_{(x-1)^2}^{-3x+13} (2 - 1) \, dy \, dx\]Solve this integral to evaluate the double integral over the region.

Verify Green's Theorem

Compare the results from the line integral and the double integral. According to Green's Theorem:\[\oint_{C} \vec{F} \cdot d\vec{r} = \iint_{R} ( \text{curl} \vec{F} ) \, dA\]If the two quantities are equal, Green's Theorem is verified.

Key concepts

Vector Field

A **vector field** is a mathematical construct commonly used in physics and engineering to represent a field of vectors in a particular region of space. Each point in the region has a vector associated with it, providing both magnitude and direction. For example, consider gravity, where at each point in space, there is a vector pointing toward the center of the Earth with a magnitude related to the strength of gravity at that point.
In the given exercise, the vector field is \(\vec{F}=\langle x+y, 2x\rangle\). Here, given any point \( (x, y) \), you can compute the corresponding vector by substituting these coordinates into the expression for \( \vec{F}\).
Vector fields are essential because they help in visualizing and understanding the behavior of dynamic systems and making calculations, such as the flow of fluid, electric fields, or, as here, evaluating line integrals.

Line Integral

The **line integral** is one of the tools used to calculate the integral of a vector field along a curve. Essentially, it measures how much a vector field \(\vec{F}\) can influence an object to move along a path \( C \).
It is like summing up the effects of the field as you travel across the path. In Green's Theorem problems, line integrals are especially useful as they allow you to deduce properties about a region's boundary based on values throughout the area.
In this exercise, we calculate two line integrals. One part follows the curve of a parabola from \( (0,1) \) to \( (3,4) \), and the second part follows a straight line back to the starting point. By evaluating these segments, you can understand how the vector field influences objects moving along this closed path.

Curl

The **curl** is a measure of the rotation of a vector field. It's a critical concept when you want to discover how much the vector field \( \vec{F} \) swirls around a given point. In two dimensions, the curl is simplified to a scalar quantity based on the differences in the partial derivatives.
For \( \vec{F}=\langle x+y, 2x \rangle \), the curl is computed as \( \frac{\partial (2x)}{\partial x} - \frac{\partial (x+y)}{\partial y} = 2 - 1 = 1 \). This means the field has a rotational effect represented by this scalar value. Understanding curl is essential because it tells us about the field's internal dynamics related to Green's Theorem, where regions' and boundaries' behavior is analyzed.

Double Integral

The **double integral** is used to calculate the accumulation of quantities over a two-dimensional area. When dealing with vector fields, it typically involves integrating the curl over a region. This process converts information about a surface into numeric values representing aspects like mass or area.
In the context of verifying Green's Theorem, the double integral of the curl over region \( R \) provides the "area-based" account to match against the "boundary-based" account of the line integral. The exercise uses the limits from \( x = 0 \) to \( x = 3 \) and visually bounded by \( y = (x-1)^2 \) and \( y = -3x + 13 \). Solving the double integral over these limits offers insights into how the field behaves across an area rather than just along a path.