Problem 24 - Essential Calculus Early Transcendentals

Question

Show that the curve with parametric equations \(x=\sin t\) \(y=\cos t, z=\sin ^{2} t\) is the curve of intersection of the surfaces \(z=x^{2}\) and \(x^{2}+y^{2}=1 .\) Use this fact to help sketch the curve.

Step-by-Step Solution

Answer

The curve is the intersection of the surfaces \(z=x^2\) and \(x^2+y^2=1\).

Step 1: Understand the Parametric Equations

The curve is defined by the parametric equations: \(x = \sin t\), \(y = \cos t\), and \(z = \sin^2 t\). This parametric representation describes a curve in three-dimensional space as a function of the parameter \(t\).

Step 2: Verify Intersection with First Surface

Substitute \(z = \sin^2 t\) into the equation of the first surface \(z = x^2\). Since \(x = \sin t\), the equation becomes \(z = \sin^2 t = x^2\). Therefore, the parametric equation satisfies the first surface.

Step 3: Verify Intersection with Second Surface

Substitute \(x = \sin t\) and \(y = \cos t\) into the equation of the second surface \(x^2 + y^2 = 1\). This becomes \(\sin^2 t + \cos^2 t = 1\), which is a trigonometric identity, confirming the parametric equations satisfy the second surface.

Step 4: Confirm Curve is Intersection of the Surfaces

Since the parametric equations satisfy both surface equations \(z = x^2\) and \(x^2 + y^2 = 1\), the curve determined by these parametric equations is indeed the intersection of these two surfaces.

Step 5: Sketch the Curve

The curve is the intersection of a cylinder \(x^2 + y^2 = 1\) (a circle along the z-axis) and a paraboloid \(z = x^2\). The curve will loop between the boundaries of the cylindrical surface with variable z values defined by the paraboloid.

Key Concepts

Surface Intersection

A surface intersection occurs when two surfaces in a space meet, forming a curve or a line. In three-dimensional geometry, understanding surface intersections can help us visualize how the surfaces interact. In this exercise, the intersection is between a cylinder and a paraboloid. The surfaces given are expressed as:

• Cylinder: \[x^2 + y^2 = 1\]This represents a circular cylinder with a radius of 1, extending infinitely along the z-axis.
• Paraboloid: \[z = x^2\]This represents a parabolic surface where the wider opening is facing the positive z-direction.

The curve of intersection is determined by the set of points that satisfy both surface equations simultaneously. The solution involves verifying how the parametric equations fit into each surface equation. Thus, it highlights that the curve is the trajectory that exists precisely where both given surfaces meet.

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that hold true for all values of the variables involved. These identities are fundamental tools in calculus and analytical geometry, making them exceedingly useful in parametric equations. A key trigonometric identity used in this particular exercise is \[\sin^2 t + \cos^2 t = 1\]This identity verifies the relationship between the sine and cosine functions.In the context of this exercise, this trigonometric identity helps to confirm that the parametric equations satisfy the cylinder equation \(x^2 + y^2 = 1\). When substituted separately into the second surface equation, it elegantly confirms that these parametric values always lie on the cylinder.

Parametric Equations

Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent parameters. In geometry, they allow the description of curves and surfaces without needing to solve y in terms of x or vice versa.For the given exercise, the parametric equations are

• \(x = \sin t\)
• \(y = \cos t\)
• \(z = \sin^2 t\)

These equations generate a three-dimensional curve, with \(t\) acting as the parameter that varies, thus tracing out the curve step by step. By substituting these equations into the surface equations, it becomes clear how the curve extends through the space defined by the intersection of these surfaces.

3D Space Visualization

Visualizing a curve in three-dimensional space can sometimes be a complex task, especially when it involves the intersection of surfaces. Here, the goal is to imagine how the curve formed by the parametric equations fits within, and around, the given surfaces.To sketch the curve:

• The surface \(x^2 + y^2 = 1\) forms a cylinder.
• The surface \(z = x^2\) forms a paraboloid.
• The intersection curve loops around the outside of the cylinder as it dips and climbs according to the paraboloid's shape.

Try visualizing the cylinder as a tube and the paraboloid as a bowl. The intersection is like a ribbon coiling around the tube, following the slope of the bowl. Using software that allows for 3D plotting can also assist greatly in providing a more tangible look at such curves, strengthening spatial reasoning.