Question

Consider the radial field $$\mathbf{F}=\langle f, g\rangle=\frac{\langle x, y\rangle}{\sqrt{x^{2}+y^{2}}}=\frac{\mathbf{r}}{|\mathbf{r}|}$$. a. Explain why the conditions of Green's Theorem do not apply to \(\mathbf{F}\) on a region that includes the origin. b. Let \(R\) be the unit disk centered at the origin and compute $$\iint_{R}\left(\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}\right) d A$$. c. Evaluate the line integral in the flux form of Green's Theorem on the boundary of \(R\). d. Do the results of parts (b) and (c) agree? Explain.

Short answer

#Answer# a. The conditions of Green's Theorem do not apply to the given vector field because it is not continuously differentiable at the origin. b. The double integral of the partial derivatives of f and g over the unit disk R centered at the origin is 2π. c. The line integral in the flux form of Green's Theorem on the boundary of R is 0. d. The results of parts (b) and (c) do not agree, which is expected because Green's Theorem does not apply to the vector field on a region that includes the origin.

Step-by-step solution

Part a: Explain why the conditions of Green's Theorem do not apply to \(\mathbf{F}\) on a region that includes the origin.

Green's Theorem states that for a positively oriented, piecewise-smooth, simple curve \(C\) in the plane and a continuously differentiable vector field \(\mathbf{F} = \langle f, g \rangle\), we have: $$\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{R} \left( \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y} \right) dA$$ where \(R\) is the region bounded by \(C\). However, this theorem requires the vector field to be continuously differentiable on \(R\) and its boundary. In the given vector field \(\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|}\), we can see that it is not defined at the origin (0,0) since the magnitude of the vector is zero at that point. Therefore, the conditions of Green's Theorem do not apply to this vector field on a region that includes the origin.

Part b: Compute the double integral over the unit disk \(R\).

We are given \(f = \frac{x}{\sqrt{x^2 + y^2}}\) and \(g = \frac{y}{\sqrt{x^2 + y^2}}\). First, we find the partial derivatives: \(\frac{\partial f}{\partial x} = \frac{y^2}{(x^2 + y^2)^{\frac{3}{2}}}\) \(\frac{\partial g}{\partial y} = \frac{x^2}{(x^2 + y^2)^{\frac{3}{2}}}\) Now, we compute the double integral over \(R\): $$\iint_{R} \left( \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} \right) dA$$ To evaluate this integral, we can convert to polar coordinates: \(x = r\cos\theta\) and \(y = r\sin\theta\) $$\iint_{R} \left(\frac{r^2\cos^2\theta + r^2\sin^2\theta}{r^3}\right) r dr d\theta$$ The limits of integration for \(r\) are from 0 to 1, and for \(\theta\) are from 0 to \(2\pi\). Therefore, the integral becomes: $$\int_{0}^{2\pi}\int_{0}^{1}\frac{r^2}{r^2} dr d\theta = \int_{0}^{2\pi}\int_{0}^{1} dr d\theta = \int_{0}^{2\pi}(\theta|_{0}^{1})d\theta = \int_{0}^{2\pi} d\theta = \left[\theta\right]_{0}^{2\pi} = 2\pi$$

Part c: Evaluate the line integral on the boundary of \(R\).

We now evaluate the line integral in the flux form of Green's Theorem on the boundary of \(R\). The boundary of \(R\) is a circle of radius 1, centered at the origin. In polar coordinates, this can be described as \(r = 1\), and \(\theta\) varies from \(0\) to \(2\pi\). We can parametrize the boundary as follows: $$\mathbf{r}(\theta) = \langle \cos\theta, \sin\theta \rangle, \quad 0 \le \theta \le 2\pi$$ The derivative of this parametrization is: $$\frac{d\mathbf{r}}{d\theta} = \langle -\sin\theta, \cos\theta \rangle$$ Now, we can evaluate the line integral: $$\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{0}^{2\pi} \frac{\mathbf{r}}{|\mathbf{r}|} \cdot \frac{d\mathbf{r}}{d\theta} d\theta = \int_{0}^{2\pi} \left\langle \frac{\cos\theta}{\sqrt{\cos^2\theta + \sin^2\theta}}, \frac{\sin\theta}{\sqrt{\cos^2\theta + \sin^2\theta}} \right\rangle \cdot \left\langle -\sin\theta, \cos\theta \right\rangle d\theta$$ Since \(\sqrt{\cos^2\theta + \sin^2\theta} = 1\), we can simplify the integral as: $$\int_{0}^{2\pi}(-\cos\theta\sin\theta + \sin\theta\cos\theta) d\theta$$ This integral evaluates to 0, as the integrand is identically zero: $$\oint_{C} \mathbf{F} \cdot d\mathbf{r} = 0$$

Part d: Do the results of parts (b) and (c) agree? Explain.

The results of parts (b) and (c) do not agree. In part (b), we computed the double integral over the unit disk \(R\) to be \(2\pi\), while in part (c), we computed the line integral on the boundary of \(R\) to be 0. This discrepancy is expected, as Green's Theorem does not apply to the given vector field \(\mathbf{F}\) on a region that includes the origin, as explained in part (a).

Key concepts

Radial Field

A radial field is a type of vector field in which each vector points directly away from or towards the origin. This is evident in the radial field given in the exercise: \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|} \). Here, \( \mathbf{r} \) is the position vector \( \langle x, y \rangle \), and the magnitude \( |\mathbf{r}| \) normalizes it. This makes the vector point in a radial direction with unit length.

Key characteristics of radial fields:

• Vectors point radially away from the origin.
• The magnitude or length of each vector in the field is uniform in the case of the normalized radial field.
• Singularities often occur at the origin, making calculations involving the origin complex or undefined.

Understanding the nature of radial fields helps in solving problems involving symmetries, like those seen in the Green's Theorem exercise.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and operations. It’s instrumental in understanding various physical systems, as it provides tools like divergence, curl, and integrals.

In the context of the exercise, Green’s Theorem, which is part of vector calculus, is used to relate a line integral around a simple closed curve to a double integral over the region it encloses.

Vector calculus allows:

• Calculation of quantities like flow, flux, and circulation in vector fields.
• Transformation of integrals across different coordinate systems, which simplifies problems.
• Analysis of conservative fields where potential functions can describe the field effectively.

Vector calculus provides a deep understanding and manipulation of physical phenomena represented by vectors.

Line Integral

A line integral is a way to integrate functions along a curve, taking into account both location and direction. In vector calculus, line integrals are used to compute work done by a force field.

For the exercise, the line integral is evaluated in the context of Green’s Theorem on the boundary of a region. The line integral is computed around a circle that forms the boundary of a unit disk.

Key aspects of line integrals:

• They combine both the magnitude of the vector field and the direction of the path.
• Useful in computing work where force fields along a path are considered.
• A line integral around a closed path (like in Green's Theorem) can reveal insights about the fields.

Knowledge of line integrals is essential to solve problems involving physical interpretations like circulation and flux.

Double Integral

Double integrals extend the concept of single integrals to functions of two variables, typically over a region within the plane. In this exercise, double integrals are used to compute the net "accumulation" of a certain quantity over an area.

For the given radial field in the exercise, transforming the integral into polar coordinates simplifies the computation. This is essential for handling circular regions like the unit disk mentioned.

Important points about double integrals:

• They allow the calculation of volume under surfaces within a specified region.
• Region boundaries can vary significantly, thus, coordinate transformation to polar, spherical, or cylindrical coordinates is often employed.
• When combined with vector calculus concepts like divergence, they provide a deeper understanding of field behavior across a surface.

Mastering double integrals enables the analysis of fields over areas, essential for advanced calculus and engineering applications.