Problem 13 - Thomas Calculus

Question

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}=4, \quad z=y$$

Step-by-Step Solution

Answer

The set of points is a helical curve on a cylinder of radius 2 with axis along the \(z\)-axis, satisfying \(z = y\).

Step 1: Analyze the first equation

The first equation is \(x^2 + y^2 = 4\). This is the equation of a circle in the \(xy\)-plane, with its center at the origin \((0, 0)\) and a radius of 2. Therefore, in 3D space, this represents a cylinder extending infinitely along the \(z\)-axis.

Step 2: Analyze the second equation

The second equation is \(z = y\). This equation implies that for any point in the space, the \(z\)-coordinate is always equal to the \(y\)-coordinate. This describes a plane in 3D space that is tilted, forming a 45-degree angle with the \(y\)-axis and the \(z\)-axis.

Step 3: Combine the constraints

To find the set of points satisfying both equations, we consider the intersection of the geometric figures described. The cylinder defined by \(x^2 + y^2 = 4\) intersects the plane \(z = y\). The intersection results in a curve where every point on the cylinder has its height (\(z\)) equal to its \(y\)-coordinate. Visually, this means slicing the cylinder with the plane to see the elliptical cross-section.

Step 4: Visualize the intersection

To visualize the intersection, consider traces of the cylinder. For each point \((x, y)\) on the circular base where \(x^2 + y^2 = 4\), assign \(z = y\). This yields a helical curve wrapping around the cylinder because the \(z\)-value follows the height attributed by its corresponding \(y\)-value on the circle.

Step 5: Provide the geometric description

The geometric description of the set of points is a helical curve inscribed on a cylindrical surface with radius 2, where the cylinder's axis is along the line \(x = 0, \; y = 0, \; z\) and every point satisfies \(z = y\). The helix spirals upwards (or downwards) along the surface, creating a tilted spiral pattern by the nature of \(z = y\).

Key Concepts

Cylinder Equation

The equation given by \(x^2 + y^2 = 4\) defines a cylinder in three-dimensional space. On the 2D plane, this equation represents a circle centered at the origin \((0, 0)\) with a radius of 2. However, when considering three-dimensional geometry, this becomes a circular cylinder extending infinitely along the \(z\)-axis.

Key characteristics of the cylinder:

• Center: Origin \((0, 0)\) in the \(xy\)-plane.
• Radius: 2.
• Infinite extension: Along the \(z\)-axis, making it a cylinder, not a circle.

In essence, you can visualize this as a circular shape drawn repeatedly at every point on the \(z\)-axis, creating a cylindrical shell. Each circle’s radius is consistent as you move up and down the \(z\)-axis.

Plane Equation

The equation \(z = y\) defines a plane in three-dimensional space. Here, each point on the plane satisfies the condition that the \(z\)-coordinate is equal to the \(y\)-coordinate.

This plane is slightly tilted, forming a 45-degree angle between both the \(y\)-axis and \(z\)-axis.

• The angle arises because changes in \(y\) directly cause exact, equal changes in \(z\). Hence, a slope of 1 on the \(y\)-\(z\) plane.

By understanding where this plane lies, you can determine its role in cutting through the cylinder. Picture the plane like a sheet that slices through space maintaining its tilt, indicating the relationship between \(z\) and \(y\).

Helical Curve Intersection

As the plane \(z = y\) intersects with the cylindrical surface described by \(x^2 + y^2 = 4\), a distinct type of curve forms: a helical curve.

The notion of a helix often brings to mind the shape of a spring or a spiral staircase. In this case, the helical curve wraps around the cylinder. This happens because the height \(z\) of any point on the cylinder is dictated by its \(y\)-coordinate, which spirals along as you trace the cylinder.

Think of it like this:

• For every position \((x, y)\) on the circle that forms the base of the cylinder, assign a corresponding \(z\) equal to \(y\).
• This creates a continuous incline around the cylinder.

Through this understanding, this helical structure can be visualized as wrapping around the cylindrical surface, making its path seem like a spiraling ascent or descent.

Three-Dimensional Geometry

Understanding three-dimensional geometry involves visualizing components like points, lines, planes, and shapes in space. Here, both the cylinder and the plane interact within this three-dimensional setting.

Always remember the following concepts:

• Planes: Infinite in two directions, they can slice through three-dimensional objects, creating intersection lines or curves.
• Cylinders: Such shapes are extended in one direction (in this case, \(z\)-axis), but maintain a consistent circular cross-section.
• Intersections: These define where two geometric figures meet, forming new shapes or paths, like the helical curve.

In essence, three-dimensional geometry allows us to break down complex forms into their fundamental elements, thereby understanding the resulting intersections and shapes. It's a matter of linking flat representations to how they manifest in 3D space.