Problem 48 - Calculus: Early Transcendental Functions

Question

Determine whether or not each region is simply-connected. (a) \(\\{(x, y): 1

Step-by-Step Solution

Answer

Region (a) \(\{(x, y): 1

Step 1: Analyzing Region (a)

The given region (a) \(\{(x, y): 1

Step 2: Analyzing Region (b)

The given region (b) \(\left{(x, y): 1

Key Concepts

Calculus

When we talk about calculus, we are dealing with the mathematical study of change, which is crucial in understanding the behavior of functions and their properties. In this context, calculus primarily helps us explore rate of change (differentiation) and accumulated change (integration).

In solving whether a region is simply-connected, calculus aids in studying and visualizing the space we're dealing with. It helps to analyze graphical representations of regions in the coordinate plane. Through the lens of calculus, we assess the conditions under which a closed loop within these regions can be smoothly contracted to a single point. This contraction process is often linked to the concept of a path integral, an integral where the function to be integrated is evaluated along a curve. In our exercise, calculus supports by visualizing the continuous shrinking of loops, critical in understanding the essence of simply-connected regions.

Inequality in Two Dimensions

Inequalities are not just about numbers on a line; they extend to two dimensions and shape the world of coordinate geometry. When we plot inequalities in two dimensions, we're defining regions of the plane that satisfy certain conditions.

For example, the inequality \(1 < x < 2\) creates a vertical strip that is infinitely wide in the y-direction, but bounded between x = 1 and x = 2. Understanding these two-dimensional inequalities allows us to visualize regions and analyze their connectivity. In a simply-connected region, any loop can be shrunk to a point without crossing the region's boundary. We assess this by looking at how the inequalities define the region and determining if there are any 'holes' or 'gaps' that would prevent such deformation.

Annulus in Mathematics

An annulus in mathematics is a ring-shaped object, a region bounded by two concentric circles with different radii. Picture a doughnut or the surface of a washer. Mathematically, an annulus is the area between the two circles, where every point within is at a distance greater than the inner radius and less than the outer radius from the center.

In our region (b), we are dealing with an annulus defined by the inequality \(1 < x^2 < 2\), which encompasses two-dimensional space but excludes the region within the inner circle, and the space outside the outer circle. Understanding annuli is crucial when studying simply-connected regions, as they inherently possess a 'hole' and therefore cannot be considered simply-connected.

Continuous Deformation

Continuous deformation is a concept that originates from topology, a field of mathematics that deals with the properties of space that are preserved under continuous transformations. Continuous deformations include stretching or bending, but not tearing or gluing.

In the language of simply-connected regions, a region is simply-connected if any loop can be continuously deformed into a single point within the region. This means there are no holes or breaks in the space. The concept of continuous deformation is integral to the understanding of the region (b) in our exercise, where despite a loop being drawn in the annulus, it cannot be shrunk to a point without leaving the region due to the hole at its center. Thus, it emphasizes the non-simply-connected nature of annuli.

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Other exercises in this chapter

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

Problem 49

Find the mass and center of mass of the region. The portion of the plane \(3 x+2 y+z=6\) inside the cylinder \(x^{2}+y^{2}=4, \rho(x, y, z)=x^{2}+1\)

View solution

Problem 49

Use the notation \(r=\langle x, y\rangle\) and \(r=\|\mathbf{r}\|=\sqrt{x^{2}+y^{2}}\) Show that \(\frac{\langle 1,1\rangle}{r}\) is not conservative.

View solution

Key Concepts

Calculus

When we talk about calculus, we are dealing with the mathematical study of change, which is crucial in understanding the behavior of functions and their properties. In this context, calculus primarily helps us explore rate of change (differentiation) and accumulated change (integration).

In solving whether a region is simply-connected, calculus aids in studying and visualizing the space we're dealing with. It helps to analyze graphical representations of regions in the coordinate plane. Through the lens of calculus, we assess the conditions under which a closed loop within these regions can be smoothly contracted to a single point. This contraction process is often linked to the concept of a path integral, an integral where the function to be integrated is evaluated along a curve. In our exercise, calculus supports by visualizing the continuous shrinking of loops, critical in understanding the essence of simply-connected regions.

Inequality in Two Dimensions

Inequalities are not just about numbers on a line; they extend to two dimensions and shape the world of coordinate geometry. When we plot inequalities in two dimensions, we're defining regions of the plane that satisfy certain conditions.

For example, the inequality \(1 < x < 2\) creates a vertical strip that is infinitely wide in the y-direction, but bounded between x = 1 and x = 2. Understanding these two-dimensional inequalities allows us to visualize regions and analyze their connectivity. In a simply-connected region, any loop can be shrunk to a point without crossing the region's boundary. We assess this by looking at how the inequalities define the region and determining if there are any 'holes' or 'gaps' that would prevent such deformation.

Annulus in Mathematics

An annulus in mathematics is a ring-shaped object, a region bounded by two concentric circles with different radii. Picture a doughnut or the surface of a washer. Mathematically, an annulus is the area between the two circles, where every point within is at a distance greater than the inner radius and less than the outer radius from the center.

In our region (b), we are dealing with an annulus defined by the inequality \(1 < x^2 < 2\), which encompasses two-dimensional space but excludes the region within the inner circle, and the space outside the outer circle. Understanding annuli is crucial when studying simply-connected regions, as they inherently possess a 'hole' and therefore cannot be considered simply-connected.

Continuous Deformation

Continuous deformation is a concept that originates from topology, a field of mathematics that deals with the properties of space that are preserved under continuous transformations. Continuous deformations include stretching or bending, but not tearing or gluing.

In the language of simply-connected regions, a region is simply-connected if any loop can be continuously deformed into a single point within the region. This means there are no holes or breaks in the space. The concept of continuous deformation is integral to the understanding of the region (b) in our exercise, where despite a loop being drawn in the annulus, it cannot be shrunk to a point without leaving the region due to the hole at its center. Thus, it emphasizes the non-simply-connected nature of annuli.