Question

Identify and sketch the following sets in spherical coordinates. $$\\{(\rho, \varphi, \theta): \rho=4 \cos \varphi, 0 \leq \varphi \leq \pi / 2\\}$$

Short answer

The given set in spherical coordinates represents a paraboloid of revolution, opening upwards and symmetric about the z-axis.

Step-by-step solution

Define Spherical Coordinates

Spherical coordinates are a three-dimensional coordinate system where a point in space is determined by three parameters: the radial distance 𝜌 from the origin, the polar angle 𝜑 (referred to as the inclination angle) from the positive z-axis, and the azimuthal angle 𝜃 from the positive x-axis in the xy-plane. In summary, the parameters of spherical coordinates are (ρ, φ, θ).

Analyze The Given Equation

We are given the equation 𝜌 = 4cosφ and the range 0≤φ≤π/2. We can see that the 𝜌 value depends on the cosine of the polar angle 𝜑, and the azimuthal angle θ is not involved in the equation. This means that for any given value of φ in the specified range, the shape should be symmetric about the z-axis. Since 0≤cos(φ)≤1 for φ in [0, π/2], the values of ρ will range from 0 to 4.

Convert to Cartesian Coordinates

To understand the shape and help us in sketching the graph, let us convert the equation from spherical coordinates to Cartesian coordinates. The conversion formulas are: x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ). Substitute 𝜌 from the given equation 𝜌 = 4cos(φ) into these formulas: x = (4cos(φ))sin(φ)cos(θ) = 2sin(2φ)cos(θ), y = (4cos(φ))sin(φ)sin(θ) = 2sin(2φ)sin(θ), z = (4cos(φ))cos(φ) = 2cos(2φ).

Identify the Shape

Now we have equations of x, y, and z in terms of φ and θ. We notice this set describes a surface where each point on the surface has x, y, and z coordinates that depend on the polar angle φ and the azimuthal angle θ. More precisely, it represents a surface of revolution, which is formed by rotating a 2D curve in the xz-plane around the z-axis. Here, the 2D curve is given by: x = 2sin(2φ), z = 2cos(2φ), for 0≤φ≤π/2. The curve in the xz-plane can be expressed as: z = ± sqrt(4-x^2). Rotating this around the z-axis, we get a paraboloid of revolution.

Sketch the Set

Now that we have identified the shape as a paraboloid of revolution, we can sketch the set in spherical coordinates. Since the set is symmetric about the z-axis, we can draw the paraboloid extending out from the origin into the first quadrant of the Cartesian coordinate system. It should look like an upward-opening paraboloid, with maximum ρ value of 4 when φ = 0 and minimum ρ value of 0 when φ = π/2. In conclusion, the given set in spherical coordinates represents a paraboloid of revolution, opening upwards and symmetric about the z-axis.

Key concepts

Coordinate Systems

In mathematics, a coordinate system is a method used to define a point in a space by a set of values, known as coordinates. Each value corresponds to a point's specific location in space. Coordinate systems help translate geometric concepts into algebraic equations, making it easier to analyze and visualize shapes and structures.

There are several types of coordinate systems, each with its unique way of defining points:

• Cartesian Coordinate System: It uses mutually perpendicular axes (x, y, z) to specify a point in space.
• Spherical Coordinate System: This system uses a radial distance (\(\rho\)), polar angle (\(\varphi\)), and azimuthal angle (\(\theta\)) to define a point.
• Cylindrical Coordinate System: It combines linear distance, angle, and height to locate a point.

Using these systems, particularly the spherical coordinates in the given exercise, simplifies the process of solving complex spatial problems by breaking them down into manageable parts.

Conversion Between Coordinate Systems

Converting between coordinate systems allows us to express points in different formats, depending on the problem's requirements. It is essential for visualizing the geometry of a space with ease and for calculations.

Spherical coordinates can be converted to Cartesian coordinates using specific formulas:

• \(x = \rho \sin(\varphi) \cos(\theta)\)
• \(y = \rho \sin(\varphi) \sin(\theta)\)
• \(z = \rho \cos(\varphi)\)

In the exercise, we started with \(\rho = 4 \cos(\varphi)\) for the spherical coordinate system. Substituting this equation into our conversion formulas allowed us to express the problem in Cartesian coordinates. Through this conversion, it was revealed that the given set actually forms part of a paraboloid in Cartesian terms, thus giving us a clearer geometric understanding.

Surface of Revolution

A surface of revolution in geometry is created when a two-dimensional curve is rotated around an axis, forming a three-dimensional shape. This process generates symmetric, smooth surfaces which are often found in physics and engineering.

In the context of this exercise, the surface of revolution is formed by revolving a curve in the \(xz\)-plane around the \(z\)-axis. The curve, given by \(x = 2\sin(2\varphi)\) and \(z = 2\cos(2\varphi)\), when revolved around the \(z\)-axis, creates a shape known as a paraboloid of revolution. This paraboloid is spiraling outwards from the origin, with its apex at the origin and extending into the first quadrant.
Understanding the concept of surfaces of revolution aids in visualizing how simple curves can turn into complex three-dimensional forms, providing a deeper insight into how shapes evolve in space.