Problem 14 - Advanced Engineering Mathematics

Question

In Problems, find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ z=x^{2}+y^{2}, z=1 ; x=\sin t $$

Step-by-Step Solution

Answer

Vector function: \( \mathbf{r}(t) = \langle \sin t, \cos t, 1 \rangle \). Curve: Circle at \( z = 1 \).

Step 1: Analyzing the given surfaces

We are given two surfaces: \( z = x^2 + y^2 \) and \( z = 1 \). These surfaces intersect at points where \( x^2 + y^2 = 1 \). This equation represents a circle in the \( xy \)-plane.

Step 2: Using the parameterization

We are also given that \( x = \sin t \). We can substitute this into the equation \( x^2 + y^2 = 1 \), which becomes \( (\sin t)^2 + y^2 = 1 \). Simplifying gives \( y^2 = 1 - \sin^2 t \).

Step 3: Solving for other variables

From the equation \( y^2 = 1 - \sin^2 t \), we know \( y = \pm \cos t \) due to the Pythagorean identity \( \sin^2 t + \cos^2 t = 1 \). We have two possible functions for \( y \): \( y = \cos t \) or \( y = -\cos t \).

Step 4: Determining the vector function

The vector function \( \mathbf{r}(t) \) describing the curve is \( \mathbf{r}(t) = \langle \sin t, y, 1 \rangle \). Using the positive solution for \( y \), we have \( \mathbf{r}(t) = \langle \sin t, \cos t, 1 \rangle \).

Step 5: Describing the curve

The parameterization \( \mathbf{r}(t) = \langle \sin t, \cos t, 1 \rangle \) traces a circle in the plane at \( z = 1 \) as \( t \) varies from \( 0 \) to \( 2\pi \).

Key Concepts

Intersection of Curves

Finding the intersection between curves is a common problem in calculus. Here, we have two surfaces: \( z = x^2 + y^2 \) and \( z = 1 \). These equations describe surfaces in three-dimensional space. To find their intersection, we set them equal: \( x^2 + y^2 = 1 \). This is our key relationship. This equation, \( x^2 + y^2 = 1 \), describes a circle in the \( xy \)-plane. Hence, the intersection of these surfaces is a curve that lies along this circle as it resides in the plane defined by \( z = 1 \).

When tackling problems with curve intersections, it’s important to think about:

• The dimensions involved (e.g., 3D space in this instance).
• The geometric shapes the equations represent (e.g., circles, planes).
• The resulting equations when comparing these shapes (like \( x^2 + y^2 = 1 \)).

Understanding curve intersection helps in visualizing how different geometric bodies relate to each other in space.

Understanding Parameterization

Parameterization is a way of expressing a curve by introducing a parameter, typically \( t \), that moves a point along the curve. In this problem, we used \( x = \sin t \) as a parameter. By substituting \( x = \sin t \) into the equation \( x^2 + y^2 = 1 \), we can express \( y \) in terms of \( t \). This use of parameterization simplifies the expression of complex curves.

Simplifying our equation, \( (\sin t)^2 + y^2 = 1 \), gives \( y^2 = 1 - \sin^2 t \). Parameterizing allows:

• Easy traversal over the curve as \( t \) changes.
• Simplifying calculations and expressions.
• Understanding the direction and speed at which the curve is traced.

As you work with parameterization, remember it is essential in describing curves or paths smoothly and comprehensively.

Pythagorean Identity in Action

The Pythagorean identity, \( \sin^2 t + \cos^2 t = 1 \), is foundational in trigonometry. In this exercise, it helps solve for \( y \). We know \( x = \sin t \), and based on our circle equation \( x^2 + y^2 = 1 \), we have \( y^2 = 1 - \sin^2 t \). The identity gives us that \( y = \pm \cos t \). This shows how trigonometric identities link to geometry by simplifying our expressions back to fundamental forms.

Some important points about the Pythagorean identity:

• It relates directly to a unit circle.
• It allows transformations between sine and cosine.
• It is a key identity in calculating lengths and angles.

Understanding this identity is critical when finding relationships in trigonometric equations, especially involving circles.

Visualizing a Circle in the xy-plane

A circle in the \( xy \)-plane is an important geometric figure whose representation greatly assists in spatial understanding. In this exercise, the equation \( x^2 + y^2 = 1 \) describes a circle of radius 1 centered at the origin. When parameterized as \( x = \sin t \) and \( y = \cos t \), it traces a circle as \( t \) ranges from 0 to \( 2\pi \), at a fixed height \( z = 1 \).

Key aspects of understanding circles include:

• The equation \( x^2 + y^2 = r^2 \) where \( r \) is the radius.
• A complete understanding of unit circles in trigonometry.
• Visualizing circular motion and pathways.

Recognizing and understanding circles when intersecting planes helps visualize many three-dimensional geometry problems efficiently.