Problem 1 - Thomas Calculus

Question

In Exercises \(1-4,\) verify the conclusion of Green's Theorem by evaluating both sides of Equations \((3)\) and \((4)\) for the field \(\mathbf{F}=M \mathbf{i}+N \mathbf{j}\) . Take the domains of integration in each case to be the disk \(R : x^{2}+y^{2} \leq a^{2}\) and its bounding circle \(C : \mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi\) $$\mathbf{F}=-y \mathbf{i}+x \mathbf{j}$$

Step-by-Step Solution

Answer

Both sides equal \( 2\pi a^2 \), verifying Green's Theorem.

Step 1: Understanding Green's Theorem

Green's Theorem states that for a continuously differentiable vector field \( \mathbf{F} = M \mathbf{i} + N \mathbf{j} \), over a region \( R \) and its boundary \( C \), the following holds: \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) \, dA \] In this problem, \( \mathbf{F} = -y \mathbf{i} + x \mathbf{j} \), and \( R \) is the disk \( x^2 + y^2 \leq a^2 \).

Step 2: Calculate the Line Integral

The vector field \( \mathbf{F} = -y \mathbf{i} + x \mathbf{j} \). The path \( C \) is parameterized as \( \mathbf{r}(t) = a \cos t \mathbf{i} + a \sin t \mathbf{j} \). Then \( d\mathbf{r} = (-a \sin t) \mathbf{i} + (a \cos t) \mathbf{j} \, dt \). \[ \mathbf{F}(\mathbf{r}(t)) = -a \sin t \mathbf{i} + a \cos t \mathbf{j} \] \[ \mathbf{F} \cdot d\mathbf{r} = (0 \cdot (-a \sin t) + x(a \cos t))(a \cos t) \, dt = a^2 \cos^2 t \, dt + a^2 \sin^2 t \, dt = a^2 \, dt \] \[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \int_0^{2\pi} a^2 \, dt = 2\pi a^2 \]

Step 3: Calculate the Double Integral

The necessary partial derivatives for Green's theorem are \( \frac{\partial N}{\partial x} = 1 \) and \( \frac{\partial M}{\partial y} = -1 \). Thus, \( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 2 \). We integrate over the region \( R: x^2 + y^2 \leq a^2 \). Convert to polar coordinates where \( x = r \cos \theta \) and \( y = r \sin \theta \), yielding \( dA = r \, dr \, d\theta \). \[ \iint_R 2 \, dA = \int_0^{2\pi} \int_0^a 2r \, dr \, d\theta \] \[ = \int_0^{2\pi} \left[ r^2 \right]_0^a \, d\theta = \int_0^{2\pi} a^2 \, d\theta = a^2 \left[ \theta \right]_0^{2\pi} = 2\pi a^2 \]

Step 4: Compare Both Integrals

Both the line integral along \( C \) and the double integral over \( R \) yielded the result \( 2\pi a^2 \). This verifies Green's Theorem for the given vector field and domain.

Key Concepts

Vector Calculus

Vector Calculus is a branch of mathematics that deals with vector fields and operations on them. It extends calculus in one-dimensional spaces to multi-dimensional spaces and involves functions that have vector components. In the context of this exercise, vector calculus provides a framework to deal with vector fields like \( \mathbf{F} = -y \mathbf{i} + x \mathbf{j} \) over specific regions in the plane.
It involves operations such as differentiation and integration of vector-valued functions. Differentiation of a vector field involves calculating derivatives of each component function. Integration, on the other hand, can be more complex due to the multidimensional nature, involving line integrals, surface integrals, and volume integrals.
Key concepts in vector calculus include gradient, divergence, and curl, each describing different aspects of change and behavior within vector fields. These mathematical tools are crucial for analyzing physical phenomena described by vector fields, providing insight into fields like fluid dynamics and electromagnetism.

Line Integral

A line integral, in vector calculus, refers to integrating a function along a curve or path. It's specifically used for scalar fields or vector fields. Here, we focus on a vector field \( \mathbf{F} \, \).
The line integral of a vector field along a curve \( C \) provides a measure of the work done by the field along that path. It is expressed as: \( \oint_C \mathbf{F} \cdot d\mathbf{r} \).
To compute this, we parameterize the path \( C \), plug this into the vector field, and solve \( \mathbf{F} \cdot d\mathbf{r} \), where \( d\mathbf{r} \) is the differential of the path parameterization. This boils down to calculating the product of the vector field and the path's tangent vector, integrated over the bounds of the parameter.

• The line integral is crucial in physics for calculating work done by forces along paths.
• This type of integral evaluates the influence a vector field has along a specific path, connecting to the concept of flux.

Double Integral

A double integral extends the concept of integration into two dimensions. It allows us to calculate the 'volume' under a surface over a region in \( 2D \).
In this exercise, we use double integrals to evaluate an area integral of the differential of a vector field over a domain \( R \). This verifies Green's Theorem which connects line and area integrals.
Double integrals require setting up the integral over a domain, defined by two variables. Conversion to suitable coordinates, like polar coordinates, often simplifies these integrals. For example, over a disk, changing to polar coordinates involving \( r \) and \( \theta \) helps simplify the limits and integrand.

• The double integral in Green’s Theorem relates to the flux or circulation of a field across a region.
• This approach emphasizes area calculations and is pivotal in fields requiring spatial data, such as engineering and physics.

Vector Field

A vector field assigns a vector to every point in space. For this exercise, we focus on the vector field \( \mathbf{F} = -y \mathbf{i} + x \mathbf{j} \), a simple yet illustrative example.
Vector fields often represent physical quantities like velocity in fluid dynamics or force fields in physics. Each vector in the field indicates direction and magnitude at that point, characterizing the system's dynamics.
In mathematics, vector fields allow the application of theorems that evaluate overall behavior. Operations like divergence and curl describe how much a field spreads out or rotates, respectively. In this problem, understanding how the field operates over a region leads to verifying Green's Theorem.

• Vector fields provide deeper insights into spatial variations and system interactions.
• Such fields underpin many real-world applications, from weather systems to electromagnetic fields.

Polar Coordinates

Polar coordinates offer an alternative to Cartesian coordinates for representing points in the plane. Here, a point is defined by a distance from a reference point and an angle from a reference direction.
Using polar coordinates is especially useful for regions with circular symmetry. For instance, the disk \( R: x^2 + y^2 \leq a^2 \), transforms elegantly with \( x = r \cos \theta \) and \( y = r \sin \theta \). This reduces complexity when integrating over circular areas.
Integrating in polar coordinates involves the differential element \( dA = r \, dr \, d\theta \), which accounts for both radius and angle. This makes solving integrals like our double integral more straightforward.

• Polar coordinates simplify certain integrals and calculations, particularly with circles and related shapes.
• This system is commonly used in areas involving radial symmetry, such as antenna patterns and wave analysis.