Problem 10 - Calculus: Early Transcendentals

Question

Suppose the curve \(z=x\) in the xz-plane is rotated around the z-axis. Find an equation for the resulting surface in spherical coordinates.

Step-by-Step Solution

Answer

The surface is a cone: \(\cos(\phi) = \sin(\phi)\).

Step 1: Understand the Rotation

Given the curve \(z = x\) in the xz-plane, we can express it parametrically as \(x(t) = t\) and \(z(t) = t\) with \(y = 0\) initially since we are in the xz-plane. Rotating around the z-axis involves a circular path for y and x about z.

Step 2: Convert to Cylindrical Coordinates

In cylindrical coordinates, the rotation means the radius \(r\) from the origin (the z-axis). The parametric equations for the circle become \(x = r\cos(\theta)\), \(y = r\sin(\theta)\), and \(z = t\). Because \(z = x = t\), and \(t = r\cos(\theta)\), we have \(z=r\cos(\theta)\).

Step 3: Transition to Spherical Coordinates

Spherical coordinates are expressed as \((\rho, \theta, \phi)\). Use the relations: \(\rho = \sqrt{x^2 + y^2 + z^2}\), \(\theta = \tan^{-1}(y/x)\), and \(z = \rho\cos(\phi)\). Substitute the parametric circle expressions: \(x=r\cos(\theta)\), \(y=r\sin(\theta)\), and equate to z.

Step 4: Solve for the Spherical Surface Equation

Using the spherical relation \(z = \rho\cos(\phi)\), substitute \(\rho = r\sec(\theta)\) and \(z = r\sec(\theta)\cos(\phi)\). We'll have \(r = \rho\sin(\phi)\), thus \(z = \rho\cos(\phi) = \rho\sin(\phi)\sec(\theta)\). Simplifying, the surface equation is \(\rho\cos(\phi) = \rho\sin(\phi)\cos(\phi)\).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a way of describing a point in space using a combination of linear and angular measurements. They help in understanding situations involving rotations, especially around an axis. Here, we represent a point using three components:

• \( r \): the radial distance from the z-axis
• \( \theta \): the angle in the xy-plane from the positive x-axis
• \( z \): the height above the xy-plane

When a curve is rotated around the z-axis, as in our original exercise, it creates a cylindrical surface. Using cylindrical coordinates helps express this in terms of radius and angle about the z-axis incredibly well.
The conversion from Cartesian coordinates \((x, y, z)\) to cylindrical involves:

• \( x = r\cos(\theta) \)
• \( y = r\sin(\theta) \)
• \( z = z \)

For our problem, after rotation, the variable \( r \) describes the fluctuating radius from the z-axis, as \( x \) and \( y \) vary while \( z = t \) remains constant.

Parametric Equations

Parametric equations are particularly useful in visualizing geometric shapes and curves. Instead of expressing \( y \) solely in terms of \( x \), they introduce an independent parameter, often \( t \), which can represent time or another varying quantity.
In the original problem, the curve \( z = x \) was expressed parametrically as \( x(t) = t \) and \( z(t) = t \), with \( y(t) = 0 \) because initially, we are observing behavior in the xz-plane.
Using parametric equations, you can rotate curves easily, as they allow changes to be described in terms of movement over time or degrees of rotation.

• \( x = r\cos(\theta) \)
• \( y = r\sin(\theta) \)
• \( z = t \)

This representation becomes invaluable for understanding curves as they rotate around an axis. As rotation occurs, \( r \) and \( \theta \) define the circle formed by repeating values of \( x \) and \( y \).

Curve Rotation

Rotating a curve around an axis transforms it into a surface. In the case of the problem, the curve \( z = x \) in the xz-plane is rotated about the z-axis. This transformation creates a three-dimensional shape, which is the shell of a surface.
During this rotation, each point on the curve sweeps out a path in the circular direction around the z-axis, turning into a full surface. By using cylindrical coordinates, we can describe this rotation mathematically as changes in \( x \) and \( y \) depend on the parameters \( r \) and \( \theta \), while \( z = t \) remains constant.
In this context, the curve rotation produces a circular cross-section in the xy-plane, forming a solid of revolution. If we track the curve's path, this movement maps out a circular path around z, generating a shape reminiscent of a vertical tube or cylinder with variable radius rods.

Surface Equation

Upon rotating the curve and transitioning through different coordinate systems, finding the surface equation is essential in characterizing the resulting shape.
The transfer to spherical coordinates involves using relations like:

• \( \rho = \sqrt{x^2 + y^2 + z^2} \)
• \( \theta = \tan^{-1}(y/x) \)
• \( z = \rho\cos(\phi) \)

Subsequent substitution and simplification fuse these with cylindrical descriptions, connecting them to spherical variables.
From the exercise, as the surfaces were transitioned to spherical coordinates, surface equations are derived:

• \( \rho\cos(\phi) = \rho\sin(\phi)\sec(\theta) \)

This gives a comprehensive representation of the created surface, incorporating both the radii and angles of cylindrical and spherical variables. Despite the complex transformations, this cohesion reveals the elegant symmetry that spherical coordinates bring to the problem.