More Direct Proof Of Andrew Pressley Exercise 1.2.3

Introduction

In the realm of differential geometry, understanding the properties of curves and surfaces is crucial for grasping more complex concepts. Andrew Pressley's Elementary Differential Geometry is a comprehensive textbook that provides a solid foundation for students and researchers alike. In this article, we will delve into Exercise 1.2.3, which involves a more direct proof of a fundamental concept in differential geometry. Our discussion will focus on the geometry and differential geometry categories.

The Problem Statement

Let $\gamma(\theta)=(r\cos(\theta),r\sin(\theta))$ where $r>0$ and $0\leq\theta<2\pi$. We are asked to prove that the curve $\gamma$ is a circle of radius $r$ centered at the origin.

The Curve $\gamma$

The curve $\gamma$ is defined by the parametric equations $x=r\cos(\theta)$ and $y=r\sin(\theta)$. To visualize this curve, we can plot the points $(x,y)$ for different values of $\theta$. As $\theta$ varies from $0$ to $2\pi$, the point $(x,y)$ traces out a circle of radius $r$ centered at the origin.

The Proof

To prove that the curve $\gamma$ is a circle of radius $r$, we need to show that the distance from the origin to any point on the curve is equal to $r$. Let's consider a point $(x,y)$ on the curve, where $x=r\cos(\theta)$ and $y=r\sin(\theta)$. The distance from the origin to this point is given by the formula:

$$d=\sqrt{x^2+y^2}$$

Substituting the expressions for $x$ and $y$, we get:

$$d=\sqrt{(r\cos(\theta))^2+(r\sin(\theta))^2}$$

Using the trigonometric identity $\sin^2(\theta)+\cos^2(\theta)=1$, we can simplify the expression for $d$:

$$d=\sqrt{r^2\cos^2(\theta)+r^2\sin^2(\theta)}$$

$$d=\sqrt{r^2(\cos^2(\theta)+\sin^2(\theta))}$$

$$d=\sqrt{r^2}$$

$$d=r$$

This shows that the distance from the origin to any point on the curve is equal to $r$, which is the definition of a circle of radius $r$ centered at the origin.

Conclusion

In this article, we provided a more direct proof of Andrew Pressley's Exercise 1.2.3, which involves a curve defined by parametric equations. We showed that the curve is a circle of radius $r$ centered at the origin by demonstrating that the distance from the origin to any point on the curve is equal to $r$. This proof is essential for understanding the properties of curves and surfaces in differential geometry.

Further Reading

For those interested in learning more about differential geometry, we recommend the following resources:

• Andrew Pressley's Elementary Differential Geometry
• Do Carmo's Differential Geometry of Curves and Surfaces
• O'Neill's Elementary Differential Geometry

These texts provide a comprehensive introduction to the subject and are highly recommended for students and researchers alike.

Glossary

• Differential geometry: The study of curves and surfaces using techniques from calculus and geometry.
• Parametric equations: Equations that describe a curve or surface in terms of a parameter.
• Circle: A set of points equidistant from a central point, called the center.
• Radius: The distance from the center of a circle to any point on the circle.

References

• Pressley, A. (2012). Elementary Differential Geometry. Springer.
• Do Carmo, M. P. (1976). Differential Geometry of Curves and Surfaces. Prentice Hall.
• O'Neill, B. (2006). Elementary Differential Geometry. Academic Press.
  Q&A: More Direct Proof of Andrew Pressley Exercise 1.2.3 =====================================================

Introduction

In our previous article, we provided a more direct proof of Andrew Pressley's Exercise 1.2.3, which involves a curve defined by parametric equations. We showed that the curve is a circle of radius $r$ centered at the origin by demonstrating that the distance from the origin to any point on the curve is equal to $r$. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the significance of this exercise in differential geometry?

A: This exercise is significant because it introduces the concept of parametric equations and shows how they can be used to describe curves and surfaces in differential geometry. It also provides a foundation for understanding more advanced topics in differential geometry, such as curvature and surface theory.

Q: What is the difference between a parametric equation and a Cartesian equation?

A: A parametric equation is an equation that describes a curve or surface in terms of a parameter, whereas a Cartesian equation is an equation that describes a curve or surface in terms of the coordinates $x$ and $y$. Parametric equations are often used to describe curves and surfaces that are not easily described by Cartesian equations.

Q: How do I know if a curve is a circle or not?

A: To determine if a curve is a circle, you can use the following criteria:

• The curve must be a closed curve, meaning that it has no beginning or end.
• The curve must be a set of points equidistant from a central point, called the center.
• The curve must have a constant radius, meaning that the distance from the center to any point on the curve is always the same.

Q: What is the relationship between the parametric equations and the Cartesian equations of a circle?

A: The parametric equations of a circle are given by $x=r\cos(\theta)$ and $y=r\sin(\theta)$, where $r$ is the radius of the circle and $\theta$ is the parameter. The Cartesian equations of a circle are given by $x^2+y^2=r^2$, where $r$ is the radius of the circle. These two sets of equations are equivalent and describe the same curve.

Q: Can I use this exercise to prove that a curve is a circle if it is not centered at the origin?

A: Yes, you can use this exercise to prove that a curve is a circle if it is not centered at the origin. Simply translate the curve so that it is centered at the origin, and then use the same proof as before to show that it is a circle.

Q: What are some common mistakes to avoid when working with parametric equations?

A: Some common mistakes to avoid when working with parametric equations include:

• Not checking if the parameter is defined for all values of the variable.
• Not checking if the parametric equations are consistent with the Cartesian equations.
• Not checking if the curve is closed or not.

Conclusion

In this article, we answered some frequently asked questions related to the more direct proof of Andrew Pressley's Exercise 1.2.3. We hope that this article has been helpful in clarifying some of the concepts and ideas involved in this exercise.

Further Reading

For those interested in learning more about differential geometry, we recommend the following resources:

• Andrew Pressley's Elementary Differential Geometry
• Do Carmo's Differential Geometry of Curves and Surfaces
• O'Neill's Elementary Differential Geometry

These texts provide a comprehensive introduction to the subject and are highly recommended for students and researchers alike.

Glossary

• Parametric equation: An equation that describes a curve or surface in terms of a parameter.
• Cartesian equation: An equation that describes a curve or surface in terms of the coordinates $x$ and $y$.
• Circle: A set of points equidistant from a central point, called the center.
• Radius: The distance from the center of a circle to any point on the circle.

References

• Pressley, A. (2012). Elementary Differential Geometry. Springer.
• Do Carmo, M. P. (1976). Differential Geometry of Curves and Surfaces. Prentice Hall.
• O'Neill, B. (2006). Elementary Differential Geometry. Academic Press.