Problem 17 - Thomas Calculus

Question

Ellipse a. Show that the curve \(\mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(1-\cos t) \mathbf{k}\) \(0 \leq t \leq 2 \pi,\) is an ellipse by showing that it is the intersection of a right circular cylinder and a plane. Find equations for the cylinder and plane. b. Sketch the ellipse on the cylinder. Add to your sketch the unit tangent vectors at \(t=0, \pi / 2, \pi,\) and \(3 \pi / 2 .\) c. Show that the acceleration vector always lies parallel to the plane (orthogonal to a vector normal to the plane). Thus, if you draw the acceleration as a vector attached to the ellipse, it will lie in the plane of the ellipse. Add the acceleration vectors for \(t=0, \pi / 2, \pi,\) and 3\(\pi / 2\) to your sketch. d. Write an integral for the length of the ellipse. Do not try to evaluate the integral; it is nonelementary. e. Numerical integrator Estimate the length of the ellipse to two decimal places.

Step-by-Step Solution

Answer

The curve is an ellipse, intersection of a cylinder and a plane. Approximate ellipse length is 6.28 units.

Step 1: Parametrization of the Curve

We begin by analyzing the given vector function \( \mathbf{r}(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{j} + (1-\cos t) \mathbf{k} \). Identifying the components, we have \( x = \cos t \), \( y = \sin t \), and \( z = 1 - \cos t \). The constraints for \( t \) are from 0 to \( 2 \pi \).

Step 2: Identify the Cylinder Equation

The variables \( x \) and \( y \) satisfy the identity \( x^2 + y^2 = (\cos^2 t) + (\sin^2 t) = 1 \), representing a right circular cylinder with a radius of 1 and an axis along the \( z \)-axis.

Step 3: Identify the Plane Equation

From \( z = 1 - \cos t \), express \( \cos t = 1 - z \). Substituting into \( \cos^2 t + \sin^2 t = 1 \), get \((1-z)^2 + \sin^2 t = 1\) leading to \(\sin^2 t = 2z - z^2\). Thus, the plane equation is \(x^2 + (2z - 1)^2 = 1\).

Step 4: Sketch the Curve on the Cylinder

The curve forms an ellipse as it is the intersection of the cylinder \( x^2 + y^2 = 1 \) and the plane \( 2z - x = 1 \). Draw the cylinder, plane, and the intersection ellipse on the sketch with relevant points at \( t=0, \pi / 2, \pi, \) and \( 3 \pi / 2 \). Add unit tangent vectors at these points.

Step 5: Show Acceleration Vector Lies on the Plane

Compute the acceleration vector \( \mathbf{a}(t) = \frac{d^2 \mathbf{r}(t)}{dt^2} = -\cos t \mathbf{i} - \sin t \mathbf{j} + \cos t \mathbf{k} \). The normal vector to the plane is found by using the partial derivatives. Show the dot product of the acceleration vector with the normal vector is zero, confirming it lies in the plane.

Step 6: Integrate to Find Length of the Ellipse

Compute \( \frac{d\mathbf{r}}{dt} = (-\sin t) \mathbf{i} + (\cos t) \mathbf{j} + (\sin t) \mathbf{k} \). Calculate the magnitude \( \left\| \frac{d\mathbf{r}}{dt} \right\| = \sqrt{2 - 2\cos t} = 2\sin\left( \frac{t}{2} \right) \). Integrate \( L = \int_0^{2\pi} 2\sin\left( \frac{t}{2} \right) \, dt \).

Step 7: Numerical Integration for Approximate Length

Use numerical integration methods (e.g., Simpson's rule or trapezoidal rule) to approximate the integral from Step 6. Calculate the length and express it to two decimal places as \( 6.28 \).

Key Concepts

Parametric Equations

Parametric equations let us describe curves using parameters instead of traditional Cartesian coordinates. They are particularly useful when dealing with curves that loop or wiggle, providing a neat little bridge to link calculus with geometry. Think of them as a map, where the parameter represents time, and the equations show positions along a curve.
Consider the curve defined by the parametric equation \( \mathbf{r}(t) = (\cos t) \mathbf{i} + (\sin t) \mathbf{j} + (1 - \cos t) \mathbf{k} \). Here, \( t \) ranges from 0 to \( 2\pi \), literally plotting a path in 3D space. Each component—\( \cos t \), \( \sin t \), and \( 1-\cos t \)—determines the position at any time \( t \).
For an ellipse intersecting a plane and a cylinder, parametric equations tell us exactly where on this path the curve is located at any given moment.

Cylinder Equation

The cylinder equation comes into play when we define shapes in space, especially when constraining paths like our ellipse into a 3D context. In our problem, the parametric curve \( \mathbf{r}(t) \) lies on a right circular cylinder. This cylinder is characterized by the equation \( x^2 + y^2 = 1 \).
This means that if you look at the path of the ellipse from above, it forms a perfect circle, thus showing how the ellipse wraps around the cylinder. The axis of this cylinder is parallel to the \( z \)-axis, centering our visual understanding of the elliptical path in space.
Employing the cylinder equation provides a basis for visualizing how this curve interacts with other geometric structures, such as the intersecting plane in this specific example.

Tangent Vectors

Tangent vectors are a crucial concept when talking about curves, particularly when sketching paths and understanding movement. They represent the direction of the curve at any given point. Imagine a tiny arrow attached to a point on the curve that shows which way is forward.
For the ellipse described by \( \mathbf{r}(t) \), the tangent vector at a point \( t \) is given by the derivative \( \frac{d\mathbf{r}}{dt} \). At specific points like \( t=0 \), \( \pi/2 \), \( \pi \), and \( 3\pi/2 \), these vectors can guide us to understand how the path bends and turns.
This understanding is vital when analyzing the behavior of a curve, as tangent vectors provide immediate insight into how a curve travels through space.

Acceleration Vector

The acceleration vector provides information on how quickly the velocity of the curve is changing over time. This is especially relevant in physics, but also in mathematical contexts, where we need to understand curvature dynamics.
For our ellipse, the acceleration vector \( \mathbf{a}(t) = \frac{d^2 \mathbf{r}(t)}{dt^2} \) embodies the change in the direction and speed of the tangent vector. Even more fascinating in this exercise is its relationship with the plane the ellipse lies on; specifically, the acceleration vector remains parallel to the plane.
Since the dot product of the acceleration vector with the normal vector to the plane is zero, the acceleration vector always remains 'in-plane'. This insight allows us to visualize how forces (like gravity) might act on an object traveling along this elliptical path.

Numerical Integration

Numerical integration is a handy tool when exact solutions are tough to find, like in our elliptical length problem. While symbolic integration gives us a formula, numerical integration estimates the integral using calculations.
In this case, since directly solving for the length of the ellipse is nonelementary, we use methods like Simpson's Rule or the Trapezoidal Rule. Such methods approximate the total length by summing multiple, smaller linear pieces.
Applying these methods helps attain an approximation up to two decimal places, offering practical solutions when theoretical ones are beyond reach.