Problem 47 - Calculus: Early Transcendental Functions

Question

Determine whether or not each region is simply-connected. (a) \(\left\\{(x, y): x^{2}+y^{2}<2\right\\}\) (b) \(\left\\{(x, y): 1

Step-by-Step Solution

Answer

(a) The region \(\left\{(x, y): x^{2}+y^{2}<2\right\}\) is simply-connected. (b) The region \(\left\{(x, y): 1

Step 1: Graphical Interpretation of the Regions

The regions of exercises (a) and (b) are defined with inequalities. By drawing these inequalities, we can represent the areas or regions graphically. (a) The equation \(x^{2}+y^{2}<2\) represents a disc with a radius less than \(\sqrt{2}\) centered at the origin. (b) The equation \(1

Step 2: Determination of Simply-Connected Properties

Based on the definitions, a simply-connected region is such that, inside it, we can always shrink any closed loop to a single point without exiting the region. The disc from part (a) can be considered simply-connected, as any closed loop within the disc can be continuously shrunk down to a single point. For part (b), the annular region is not simply-connected. Any loop that encircles the hole in the center of the annulus cannot be shrunk to a point without exiting the region, since it cannot pass through the inaccessible region in the center.

Key Concepts

Graphical Interpretation

Understanding the visual representation of mathematical regions helps us better comprehend the concept of simply-connected regions. When we graph the equations given in an exercise, we translate abstract inequalities into concrete visual forms.

For example, consider the equations from the problem:

• \(x^{2}+y^{2}<2\) describes a circle's interior, not including the boundary. This inequality represents a disc centered at the origin with a radius of \(\sqrt{2}\).
• \((1

By utilizing graphical interpretation, these abstract concepts become clearer and accessible. Each region’s characteristics are echoed in their visual representation, serving as a foundational tool for further analysis.

Disc and Annular Regions

Disc and annular regions both stem from circular equations but differ significantly in their structure and properties. A disc is a simple, solid area contained entirely within a single circle. It encompasses all the points less than a radial distance away from the center point.

Conversely, an annular region is essentially a ring, defined by two circles. It includes all the points between these two boundaries, excluding the inner circle’s space. For instance, in our problem:

• A disc is exemplified by \(x^{2}+y^{2}<2\), a continuous region entirely contained within a single boundary.
• An annular region, given by \(1

The differences in structure impact whether these regions are simply-connected, affecting how loops interact within these spaces.

Mathematical Inequality Representation

Inequalities in mathematics represent regions or areas on a graph. When we say an inequality such as \(x^{2}+y^{2}<2\), it specifies all the \(x, y\) coordinate points that satisfy this condition, forming a disc. This expression uses the inequality symbol to exclude its boundary (a circle with radius \(\sqrt{2}\)).

For the inequalities seen in annular regions, like \(1

\(x^{2}+y^{2}<2\) is a simple inequality representing a bounded disc.

\(1
Understanding these representations allows solving problems involving such geometric spaces and grasping the complexity and simplicity of their boundaries.

Closed Loop Analysis

The concept of simply-connected regions relies heavily on the ability to manipulate closed loops within the region. In essence, a region is simply-connected if a loop can be shrunk to a point without leaving the confines of the region.

For example, within a disc defined by \(x^{2}+y^{2}<2\), any loop drawn can gradually contract into its center without touching the boundary. This makes the disc properly simply-connected.

On the other hand, the annular region defined by \(1

Discs allow loops to contract fully, aligning with simply-connected properties.

Annular regions cannot permit certain loops to contract without `jumping` out of the region, indicating complexity in connectivity.

Thus, closed loop analysis serves as a pivotal method to evaluate and distinguish the connectivity nature of a region.

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Problem 47

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Problem 48

Other exercises in this chapter

Problem 47

Use the notation \(r=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}\) $$\text { Find } \nabla\left(r^{3}\right)$$

View solution

Problem 47

If \(f\) is a scalar function and \(\mathbf{F}\) a vector field, show that $$\nabla \cdot(f \mathbf{F})=\nabla f \cdot \mathbf{F}+f(\nabla \cdot \mathbf{F})$$

View solution

Problem 48

Evaluate the flux integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\) \(\mathbf{F}=\langle 3, z, y\rangle, S\) is the boundary of the region between \(z=8-2

View solution

Problem 48

If \(f\) is a scalar function and \(\mathbf{F}\) a vector field, show that $$\nabla \times(f \mathbf{F})=\nabla f \times \mathbf{F}+f(\nabla \times \mathbf{F})$

View solution

Key Concepts

Graphical Interpretation

Understanding the visual representation of mathematical regions helps us better comprehend the concept of simply-connected regions. When we graph the equations given in an exercise, we translate abstract inequalities into concrete visual forms.

For example, consider the equations from the problem:

• \(x^{2}+y^{2}<2\) describes a circle's interior, not including the boundary. This inequality represents a disc centered at the origin with a radius of \(\sqrt{2}\).
• \((1

By utilizing graphical interpretation, these abstract concepts become clearer and accessible. Each region’s characteristics are echoed in their visual representation, serving as a foundational tool for further analysis.

Disc and Annular Regions

Disc and annular regions both stem from circular equations but differ significantly in their structure and properties. A disc is a simple, solid area contained entirely within a single circle. It encompasses all the points less than a radial distance away from the center point.

Conversely, an annular region is essentially a ring, defined by two circles. It includes all the points between these two boundaries, excluding the inner circle’s space. For instance, in our problem:

• A disc is exemplified by \(x^{2}+y^{2}<2\), a continuous region entirely contained within a single boundary.
• An annular region, given by \(1

The differences in structure impact whether these regions are simply-connected, affecting how loops interact within these spaces.

Mathematical Inequality Representation

Inequalities in mathematics represent regions or areas on a graph. When we say an inequality such as \(x^{2}+y^{2}<2\), it specifies all the \(x, y\) coordinate points that satisfy this condition, forming a disc. This expression uses the inequality symbol to exclude its boundary (a circle with radius \(\sqrt{2}\)).

For the inequalities seen in annular regions, like \(1

\(x^{2}+y^{2}<2\) is a simple inequality representing a bounded disc.

\(1
Understanding these representations allows solving problems involving such geometric spaces and grasping the complexity and simplicity of their boundaries.

Closed Loop Analysis

The concept of simply-connected regions relies heavily on the ability to manipulate closed loops within the region. In essence, a region is simply-connected if a loop can be shrunk to a point without leaving the confines of the region.

For example, within a disc defined by \(x^{2}+y^{2}<2\), any loop drawn can gradually contract into its center without touching the boundary. This makes the disc properly simply-connected.

On the other hand, the annular region defined by \(1

Discs allow loops to contract fully, aligning with simply-connected properties.

Annular regions cannot permit certain loops to contract without `jumping` out of the region, indicating complexity in connectivity.

Thus, closed loop analysis serves as a pivotal method to evaluate and distinguish the connectivity nature of a region.