Problem 13 - 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. $$ x^{2}+y^{2}=9, z=9-x^{2} ; x=3 \cos t $$

Step-by-Step Solution

Answer

The vector function is \(\mathbf{r}(t) = \langle 3 \cos t, 3 \sin t, 9 \sin^2 t \rangle\).

Step 1: Identify the Parameters and Equations

We are given two surfaces: the first is a cylinder defined by the equation \(x^2 + y^2 = 9\), and the second is a paraboloid defined by \(z = 9 - x^2\). The parameter provided is \(x = 3 \cos t\). We need to express \(y\) and \(z\) using this parameter \(t\).

Step 2: Express \(y\) in terms of \(t\)

Since \(x = 3 \cos t\), substitute \(x\) into the equation \(x^2 + y^2 = 9\):\[(3 \cos t)^2 + y^2 = 9\]Which simplifies to:\[9 \cos^2 t + y^2 = 9\]Isolating \(y^2\) gives:\[y^2 = 9 - 9 \cos^2 t\]Since \(1 - \cos^2 t = \sin^2 t\), substitute to find \(y\):\[y = 3 \sin t\]

Step 3: Express \(z\) in terms of \(t\)

Substitute \(x = 3 \cos t\) into \(z = 9 - x^2\):\[z = 9 - (3 \cos t)^2\]This simplifies to:\[z = 9 - 9 \cos^2 t\]Rewriting using the trigonometric identity for \(\sin^2 t\):\[z = 9 \sin^2 t\]

Step 4: Construct the Vector Function

Using the parameter \(t\), the vector function that describes the curve \(C\) is:\[\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle = \langle 3 \cos t, 3 \sin t, 9 \sin^2 t \rangle\]

Step 5: Sketch the Curve

The curve \(C\) is a circular path in the \(xy\)-plane with radius 3 due to \(x = 3 \cos t\) and \(y = 3 \sin t\), combined with the function \(z = 9 \sin^2 t\) that adds vertical movement dependent on \(y\). This results in a sinusoidal variation in \(z\) along the circular path.

Key Concepts

Curve Intersection

When analyzing mathematical problems involving multiple surfaces, finding where these surfaces intersect is crucial. The intersection of surfaces is often described by a "curve intersection". Imagine two surfaces in three-dimensional space. Where they meet, they form a curve. The challenge lies in expressing this curve mathematically. In the given exercise, we determined the intersection between a cylinder and a paraboloid. Here's how it works:

• We have a cylinder defined by the equation \(x^2 + y^2 = 9\), indicating a circular cross-section in the plane.
• We also have a paraboloid described by \(z = 9 - x^2\), a surface curving downwards as \(x\) increases.

The intersection, described as a curve, combines elements from both surfaces. The focus is on expressing this curve with respect to a certain parameter, which simplifies the understanding of its trajectory in space.

Parametric Equations

Parametric equations are incredibly useful when describing complex curves. Rather than expressing \(y\) directly in terms of \(x\) or \(z\), we use a common variable, the parameter, often denoted as \(t\). In the exercise, we worked with the parameter \(x = 3 \cos t\) to express both \(y\) and \(z\):

• Start with the known parameter \(x = 3 \cos t\).
• Substitute in the cylindrical equation: \(y = 3 \sin t\).
• Then, substitute into the paraboloid equation: \(z = 9 \sin^2 t\).

By doing this, we can describe every point on the curve with a single parameter \(t\), resulting in the vector function \(\mathbf{r}(t) = \langle 3 \cos t, 3 \sin t, 9 \sin^2 t \rangle\). This representation is compact and provides insights into how the curve moves in space as \(t\) changes.

Cylinders and Paraboloids

Understanding the characteristics of basic surfaces like cylinders and paraboloids can help tremendously in sketching curves of intersections.A **cylinder** in this context is a surface with a circular cross-section. The equation \(x^2 + y^2 = 9\) represents a cylinder extended infinitely along the \(z\)-axis. Every point on this cylinder satisfies this equation, resulting in a circular pattern when cut by horizontal planes. A **paraboloid**, described here by \(z = 9 - x^2\), is a surface that opens or closes as a parabola. Essentially, it resembles a bowl, curving downwards. The intersection of these two surfaces shapes a 3D pathway that can be described using parametric equations. By understanding each surface's inherent geometry, it becomes easier to predict and visualize how they intertwine and the nature of the curve they form.

Trigonometric Identities

Trigonometric identities, especially those involving sine and cosine, play a pivotal role in simplifying complex mathematical expressions. In this exercise, these identities are key to expressing components of the intersection curve.Consider the identity \(1 - \cos^2 t = \sin^2 t\). It is derived from the Pythagorean identity and is used repeatedly:

• To find \(y\), starting from \(9 - 9 \cos^2 t\), recognize it simplifies to \(9 \sin^2 t\). Thus, \(y = 3 \sin t\).
• Similarly, when calculating \(z\), \(9 - 9 \cos^2 t\) yields \(z = 9 \sin^2 t\).

These identities allow us to elegantly express the components of the vector function. It shows how trigonometric identities not only simplify calculations but also help in capturing the essence of curves formed by intersecting surfaces.