Question

a. Show that a torus with radii \(R>r\) (see figure) may be described parametrically by \(r(u, v)=\langle(R+r \cos u) \cos v,(R+r \cos u) \sin v, r \sin u\rangle\) for \(0 \leq u \leq 2 \pi, 0 \leq v \leq 2 \pi\) b. Show that the surface area of the torus is \(4 \pi^{2} R r\).

Short answer

Question: Show that the given parametric representation describes a torus with radii R>r and find its surface area. Answer: The parametric representation r(u, v) does indeed describe a torus with radii R and r, as confirmed by the analysis of the coordinate transformations. The surface area of the torus is 4π² * R * r.

Step-by-step solution

Understand the torus and its parametric representation

A torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. Here, R represents the distance from the center of the circle being revolved to the axis of revolution, and r is the radius of the circle. The parametric representation uses two parameters, u and v, to define each point on the torus.

Verify the torus representation

To verify that r(u, v) indeed represents a torus, we can analyze the coordinate transformations. For (x, y, z) coordinates, we get: x = (R + r*cos(u))*cos(v) y = (R + r*cos(u))*sin(v) z = r*sin(u) Notice that: 1. For a fixed value of u, (x, y, z) will resemble the equation for a circle with radius r*cos(u). 2. As v varies from 0 to 2π, x and y will trace out a circle with radius R+r*cos(u), meaning that the smaller circle is being revolved around the z-axis, creating a torus. These observations confirm that the representation r(u, v) indeed describes a torus with the given radii R and r.

Find the surface area of the torus

To find the surface area of the torus, we need to calculate the differential dS and integrate over the given range of parameters. The differential dS can be computed by finding the cross product of the partial derivatives of the position vector: dS = |∂r/∂u x ∂r/∂v| dudv Compute the partial derivatives: ∂r/∂u = <-r * sin(u) * cos(v), -r * sin(u) * sin(v), r * cos(u)> ∂r/∂v = <-(R + r * cos(u)) * sin(v), (R + r * cos(u)) * cos(v), 0> Compute the cross product: ∂r/∂u x ∂r/∂v = Find the magnitude of the cross product: |∂r/∂u x ∂r/∂v| = √((r² * cos²(u) * cos²(v)) + (r² * cos²(u) * sin²(v)) + r² * (R + r * cos(u))²) = √((r² * cos²(u) * (cos²(v) + sin²(v))) + r² * (R + r * cos(u))²) = r(R + r*cos(u)) Now, integrate dS with respect to u and v over the given range: Surface Area = ∫(∫(r * |(R + r * cos(u))|, u=0 to 2π), v=0 to 2π) = ∫(∫(r * (R + r * cos(u)), u=0 to 2π), v=0 to 2π) Using the identity cos²(u) + sin²(u) = 1, we can change the variables and compute the integral: Surface Area = r∫(∫(R + r * cos(u), u=0 to 2π), v=0 to 2π) = r * (2π) * ∫(R + r * cos(u), u=0 to 2π) = 4π² * R * r * (1 - cos(0) + cos(2π)) = 4π² * R * r Thus, the surface area of the torus is 4π² * R * r.

Key concepts

Surface of Revolution

The concept of a surface of revolution is fundamental when visualizing and defining three-dimensional shapes such as spheres, cylinders, and tori. A torus, for example, is created by revolving a circle around an axis that is coplanar with the circle but does not intersect it. The circle that creates the torus is known as the generating circle, while the line it revolves around is the axis of revolution.

For a torus with a generating circle radius of r and a distance R from the axis of revolution to the center of the generating circle, every point on the torus's surface maintains a consistent relationship to this axis. By using sinusoidal functions, we can describe the position of any point on the torus in terms of two angles, commonly referred to as u and v, which correspond to different rotations around the axis.

Surface Area Integration

Computing the surface area of complex shapes like a torus requires calculus, specifically surface area integration. When we break down the surface of a torus into infinitesimally small patches, each can be approximated as a rectangle with sides determined by differential changes in u and v. To find the surface area of the entire torus, we sum up the areas of these tiny patches over the entire range of u and v, which is known as integrating over the surface.

The differential element of surface area, dS, is one such patch, and is calculated using the magnitudes of the vectors that define its sides. These vectors are the partial derivatives of the position function with respect to u and v. By integrating this differential element over the full range of our parameters—0 to 2π for both—we calculate the total surface area of the torus.

Partial Derivatives

Partial derivatives are crucial in many fields such as physics, engineering, and economics because they allow us to examine how a multivariable function changes as we alter just one of the variables while holding the others constant. In the context of a torus, the position vector r(u, v) is a function of two variables: u and v.

The partial derivative of this function with respect to u—notated as ∂r/∂u—gives us a vector that points in the direction of the greatest rate of increase of the position function as only u changes. Similarly, the partial derivative with respect to v, ∂r/∂v, indicates the direction of increase as only v changes. These derivatives are the sides of an infinitesimal patch on the torus's surface, and their cross product gives us the area of that patch.

Cross Product in Vector Calculus

In vector calculus, the cross product is a binary operation on two vectors in three-dimensional space. It produces a third vector that is perpendicular to the plane containing the original two vectors. This is critical when determining areas in three dimensions.

To find the area of a parallelogram defined by two vectors, we take their cross product and then calculate its magnitude. For the computations related to the torus, the cross product of the partial derivatives ∂r/∂u and ∂r/∂v gives us a vector whose magnitude represents the area of the differential patch. That's why the magnitude of the cross product plays a central role in calculating the surface area of the torus, as we integrate this magnitude over the entire range of our parameters.