Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the vector forms of Green's Theorem to calculate (a) \(\oint_{C} \mathbf{F} \cdot \mathbf{n} d s\) and (b) \(\oint_{C} \mathbf{F} \cdot \mathbf{T} d s\). \(\mathbf{F}=y^{3} \mathbf{i}+x^{3} \mathbf{j} ; C\) is the unit circle.

Step-by-Step Solution

Answer

Both integrals evaluate to zero.

Step 1: Understand the Problem

We need to use Green's Theorem in its two vector forms to evaluate two line integrals, one normal (a) and one tangential (b) to circle \( C \), which is the unit circle centered at the origin. The vector field given is \( \mathbf{F} = y^3 \mathbf{i} + x^3 \mathbf{j} \).

Step 2: Parameterize the Unit Circle

The unit circle \( C \) can be parameterized using \( \mathbf{r}(t) = \cos t \, \mathbf{i} + \sin t \, \mathbf{j} \), where \( t \) ranges from \( 0 \) to \( 2 \pi \). The differential arc length element is \( d\mathbf{s} = \frac{d\mathbf{r}}{dt} dt = (-\sin t \, \mathbf{i} + \cos t \, \mathbf{j}) \, dt \).

Step 3: Establish 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 two vector forms of Green's Theorem are\[ \oint_{C} \mathbf{F} \cdot \mathbf{T} \, ds = \iint_{R} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA \]and\[ \oint_{C} \mathbf{F} \cdot \mathbf{n} \, ds = \iint_{R} \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \right) \, dA, \]where \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} \).

Step 4: Determine the Partial Derivatives

Identify \( P = y^3 \) and \( Q = x^3 \). We have:\( \frac{\partial P}{\partial y} = 3y^2 \), \( \frac{\partial Q}{\partial x} = 3x^2 \), and they lead to\( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3x^2 - 3y^2 \) and \( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 0 \) as \( \frac{\partial P}{\partial x} = 0 \) and \( \frac{\partial Q}{\partial y} = 0 \).

Step 5: Integrate to Find (a) \( \oint_{C} \mathbf{F} \cdot \mathbf{n} \, ds \)

Using the information that \( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} = 0 \), the double integral becomes:\[ \iint_{R} 0 \, dA = 0. \]Thus, \( \oint_{C} \mathbf{F} \cdot \mathbf{n} \, ds = 0 \).

Step 6: Integrate to Find (b) \( \oint_{C} \mathbf{F} \cdot \mathbf{T} \, ds \)

From earlier, we have\( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3x^2 - 3y^2 \). Over the unit disk, this integral evaluates as:\[ \iint_{R} (3x^2 - 3y^2) \, dA = 0 \] using symmetry of squares around \( x \) and \( y \). Thus, \( \oint_{C} \mathbf{F} \cdot \mathbf{T} \, ds = 0 \).

Key Concepts

Line Integrals

Line integrals are a special type of integral where we evaluate a function along a curve. Imagine tracing a path along a shape, like walking along the edge of a circle. We want to understand how a certain function behaves while moving along this path. In the context of Green's Theorem, line integrals help us relate what happens on the path (\(C\), the unit circle) to what happens inside the bounded area.

There are typically two types of line integrals:

• Scalar line integrals, which involve integrating a scalar field along a curve.
• Vector line integrals, which are often used with vector fields to evaluate the effect of the field along a path or curve.

In our Green's Theorem application, we are analyzing vector integrals for their tangential and normal components along the unit circle.

Vector Fields

A vector field assigns a vector to every point in a region of space. Picture the way wind flows over a landscape; each point in the area has a speed and direction. In mathematical terms, a vector field can be written as \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors in the x and y directions, respectively.

For our exercise, the vector field is \(\mathbf{F} = y^3 \mathbf{i} + x^3 \mathbf{j}\). This tells us that each point's vector involves components that arise from cubic expressions of the coordinates. Vector fields are essential in physics and engineering because they describe the forces acting at every point in a field, like electromagnetic forces or fluid dynamics.

• Understanding vector fields is crucial for applying the correct integrations and transformations in physics.
• They visually represent complex equations and help in comprehending the flow and strength of forces or quantities across a region.

Unit Circle

The unit circle is a critical concept in mathematics, notably within trigonometry and calculus. Simply put, it's a circle with a radius of one unit. It is an essential tool because it provides a straightforward way to describe oscillations, rotations, and many physical phenomena.

In our context, the unit circle is parameterized using \(\mathbf{r}(t) = \cos t \mathbf{i} + \sin t \mathbf{j}\), where \(t\) ranges from 0 to \(2\pi\). This parameterization describes a complete counterclockwise journey around the circle. The beauty of the unit circle lies in its symmetry, which frequently simplifies many mathematical calculations, such as integrals involving trigonometric functions because of the even spread of the cosine and sine values over one complete cycle.

• It serves as an excellent reference structure for understanding waves and circular motion.
• It simplifies the process of converting between angular and linear measurements.

Partial Derivatives

Partial derivatives measure how a function changes as its variables change, focusing on one variable at a time. They are a fundamental concept in multivariable calculus and essential for understanding how functions vary in higher dimensions.

For our exercise with Green's Theorem, we have two components of the vector field: \(P = y^3\) and \(Q = x^3\). The partial derivative \(\frac{\partial P}{\partial y} = 3y^2\) tells us how \(P\) changes with respect to \(y\). Similarly, \(\frac{\partial Q}{\partial x} = 3x^2\) represents the rate of change of \(Q\) with \(x\). When applying Green's Theorem, these derivatives allow us to conduct the conversion from line integrals over a path to double integrals over the area bounded by the path.

• Partial derivatives are the building blocks for differential equations that describe real-world phenomena.
• They're pivotal for optimization problems, where we seek to find maximum or minimum values of functions subject to constraints.