Question

Verify that the curve \(\mathbf{r}(t)\) lies on the given surface. Give the name of the surface. $$\begin{aligned}&\mathbf{r}(t)=\langle(3+\cos 15 t) \cos t,(3+\cos 15 t) \sin t, \sin 15 t\rangle\\\&(3-\sqrt{x^{2}+y^{2}})^{2}+z^{2}=1 \text { (Hint: See Example 4.) \end{aligned}$

Short answer

Answer: The curve 𝑟(𝑡) lies on a right circular cylinder with the center being the z-axis and radius equals to 1.

Step-by-step solution

Identify the components of the curve vector 𝑟(𝑡)

The curve vector 𝑟(𝑡) is given by: $$ \mathbf{r}(t) = \langle (3 + \cos 15t) \cos t, (3 + \cos 15t) \sin t, \sin 15t \rangle $$ So we have: $$ x = (3 + \cos 15t) \cos t \ , y = (3 + \cos 15t) \sin t \ , z = \sin 15t $$

Find the equation relating x and y

Since we have: $$ x = (3 + \cos 15t) \cos t, y = (3 + \cos 15t) \sin t $$ Divide y by x: $$ \frac{y}{x}=\frac{(3 + \cos 15t) \sin t}{(3 + \cos 15t) \cos t}=\frac{\sin t}{\cos t}=\tan t $$ Now, we can express x in terms of y: $$ x = (3 + \cos 15t) \cos t = y \cot t = \frac{y}{\tan t} $$

Substitute x, y, and z into the surface equation

Now substitute x, y, and z into the surface equation: $$ (3 - \sqrt{x^2 + y^2})^2 + z^2 = 1 \tag{1} $$ We have x in terms of y, and z in terms of t: $$ x = \frac{y}{\tan t} \quad , \quad z = \sin 15t $$ Substitute these into equation (1): $$ \left(3 - \sqrt{\left(\frac{y}{\tan t}\right)^2 + y^2}\right)^2 + (\sin 15t)^2 = 1 $$ Simplify the equation: $$ \left(3 - \sqrt{\frac{y^2}{\tan^2 t} + y^2}\right)^2 + (\sin^2 15t) = 1 \tag{2} $$

Verify the surface equation is true

Recall the curve vector 𝑟(𝑡): $$ \mathbf{r}(t) = \langle (3 + \cos 15t) \cos t, (3 + \cos 15t) \sin t, \sin 15t \rangle $$ Use the equation for x and y we found in step 2, then equation (2) becomes: $$ \left(3 - \sqrt{x^2 + y^2}\right)^2 + z^2 = (3 - (3 + \cos 15t))^2 + (\sin^2 15t) = 1 $$ Simplify: $$ (\cos^2 15t) + (\sin^2 15t) = 1 $$ This equation is true since it's an identity in trigonometry, so the curve 𝑟(𝑡) lies on the given surface.

Name the surface

The given surface equation is: $$ (3 - \sqrt{x^2 + y^2})^2 + z^2 = 1 $$ This equation is a surface of revolution about the z-axis. We can rewrite the equation in cylindrical coordinates (ρ, θ, z), where ρ = √(x^2 + y^2): $$ (3 - ρ)^2 + z^2 = 1 $$ This surface is called a right circular cylinder with the center being the z-axis and radius equals to 1.

Key concepts

Surface Verification

Surface verification involves showing that a curve lies on a specific surface. For the given curve \(\mathbf{r}(t) = \langle(3+\cos 15t)\cos t, (3+\cos 15t)\sin t, \sin 15t\rangle\), we need to verify that it lies on the surface defined by \((3-\sqrt{x^2+y^2})^2+z^2=1\). To accomplish this, we perform the following steps:

• Identify the individual components of the parametric curve to get values for \(x\), \(y\), and \(z\).
• Substitute the expressions of \(x\), \(y\), and \(z\) back into the surface equation.
• Upon substitution, simplify and check if the simplified equation holds true, which would confirm that the curve lies on the surface.

Performing these steps ensures that the parametric curve is indeed part of the surface equation provided. In our example, through substitution and simplification, the equation simplifies correctly to show it aligns with the provided surface equation, confirming that the curve \(\mathbf{r}(t)\) does indeed lie on the given surface.

Trigonometric Identity

A trigonometric identity involves equations that are true for all values of the variables involved. In this exercise, we use the trigonometric identity \(\cos^2 15t + \sin^2 15t = 1\) to verify the curve lies on the surface.Trigonometric identities help simplify complex equations and are a powerful tool in demonstrating relationships between trigonometric functions. Specifically, the identity \(\cos^2 \theta + \sin^2 \theta = 1\) is a fundamental one used in various applications in mathematics and physics.Being familiar with these identities, such as the Pythagorean identities (like the one used here), can significantly ease calculations involving periodic functions and can aid in solving problems involving oscillation, rotation, and other related physical concepts.

Surface of Revolution

A surface of revolution is created by rotating a curve around an axis. The given equation \((3 - \sqrt{x^2 + y^2})^2 + z^2 = 1\) describes a surface of revolution about the z-axis. It can be better understood when analyzed using cylindrical coordinates, where we visualize \(x\) and \(y\) in terms of a single distance \(\rho = \sqrt{x^2 + y^2}\).The process of understanding this surface involves evaluating how the distance from the axis of rotation changes along the height. In our exercise, the transformation shows a right circular cylinder.

• This cylinder is centered on the z-axis.
• Its radius is effectively the constant shift due to the rotation terms \(3 - \rho\).
• The presence of \(z^2 = 1\) further confirms that our surface extends uniformly as a circular cross-section along the revolution axis.

The beauty of surfaces of revolution is their symmetry and uniformity, which results from the consistent nature of rotation. Understanding these surfaces lays a foundation for more complex geometrical constructs used in fields like engineering and physics.