Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

Make the required change in the given equation. \(r^{2}+2 z^{2}=4\) to spherical coordinates

Step-by-Step Solution

Answer

\(\rho^{2}(1 + \cos^{2}(\phi)) = 4\) in spherical coordinates.

Step 1: Understanding Spherical Coordinates

In spherical coordinates, a point in space is represented by three parameters: \(\rho\) (the radial distance from the origin), \(\theta\) (the azimuthal angle in the \(xy\)-plane from the positive x-axis), and \(\phi\) (the polar angle from the positive z-axis).

Step 2: Recognize the Variables to Convert

The given equation is expressed in terms of \(r\) and \(z\), which are cylindrical coordinates. We need to convert \(r\) and \(z\) to their spherical equivalents. In spherical coordinates, \(r = \rho \sin(\phi)\) and \(z = \rho \cos(\phi)\).

Step 3: Substitute Spherical Values

Substitute the spherical equivalents into the given equation: \(r^{2} + 2z^{2} = 4\) becomes \((\rho \sin(\phi))^{2} + 2(\rho \cos(\phi))^{2} = 4\).

Step 4: Expand and Simplify the Equation

Now, expand the terms. We have:\[(\rho^{2} \sin^{2}(\phi)) + 2(\rho^{2} \cos^{2}(\phi)) = 4\]Factor out \(\rho^{2}\):\[\rho^{2}(\sin^{2}(\phi) + 2\cos^{2}(\phi)) = 4\].

Step 5: Use a Trigonometric Identity

Identify and use the identity \(\sin^{2}(\phi) + \cos^{2}(\phi) = 1\). Substitute this into \(\sin^2(\phi) + 2\cos^2(\phi)\):\[\rho^{2}(1 + \cos^{2}(\phi)) = 4\].

Step 6: Simplify the Resulting Expression

Now, the equation simplifies to: \[\rho^{2}(1 + cos^{2}(\phi)) = 4\]. This is the equation in spherical coordinates.

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are a clever way to extend polar coordinates into three dimensions. They're especially useful when working with problems involving circular symmetry along a given axis, like tubes, pipes, or cylindrical shapes. Instead of the usual \(x, y, z\) Cartesian coordinates, cylindrical coordinates use the symbols \((r, \theta, z)\). Here’s a breakdown:

• \(r\): The radial distance from the origin to the projection of the point in the \(xy\)-plane. It is akin to the radius in polar coordinates.
• \(\theta\): The angle between the positive x-axis and the line connecting the origin to the projection of the point, measured in the \(xy\)-plane. It’s how much you rotate around the z-axis.
• \(z\): The height above the \(xy\)-plane, which is the same as in Cartesian coordinates.

This system offers a more intuitive representation when working with rotational symmetry and vertical distances. For example, when a point is at \(r=2, \theta=\pi/4, z=3\), it means you should move 2 units away from the origin in the radial direction, rotate \(\pi/4\) radians counter-clockwise from the positive x-axis, and move 3 units up vertically.

Trigonometric Identity

Trigonometric identities are essential tools for simplifying expressions and solving equations involving angles. They are formulas passed down over centuries that relate the trigonometric functions to one another. A popular and frequently used identity is \(\sin^{2}(\phi) + \cos^{2}(\phi) = 1\).
This particular identity, known as the Pythagorean identity, expresses a fundamental property of the trigonometric functions. Whenever you encounter an equation with both \(\sin\) and \(\cos\) components, this identity can seamlessly simplify the expression.
In practice, understanding these identities can make coordinate conversions and transformations much more straightforward. For instance, if you see a situation where you have an expression like \(\sin^{2}(\phi) + k\cos^{2}(\phi)\), recognizing that \(\sin^{2}(\phi) + \cos^{2}(\phi) = 1\) allows you to substitute and simplify. For the given exercise, it helped translate \(\rho^{2}(\sin^{2}(\phi) + 2\cos^{2}(\phi)) = 4\) into \(\rho^{2}(1 + \cos^{2}(\phi)) = 4\), thus simplifying the expression into a more workable form.

Coordinate Conversion

Coordinate conversion involves translating points from one coordinate system to another. It's particularly valuable when the geometry or symmetry of a problem is better understood in one system.

• Spherical to Cartesian: The spherical coordinates \((\rho, \theta, \phi)\) are converted as follows:
  • \(x = \rho \sin(\phi) \cos(\theta)\)
  • \(y = \rho \sin(\phi) \sin(\theta)\)
  • \(z = \rho \cos(\phi)\)
• Cylindrical to Spherical: Conversely, you can express cylindrical coordinates \((r, \theta, z)\) in terms of spherical coordinates with:
  • \(\rho = \sqrt{r^2 + z^2}\)
  • \(\phi = \arccos\left(\frac{z}{\rho}\right)\)
  • \(\theta = \theta\) (unchanged)

The process might seem challenging at first, but practicing these transformations helps facilitate problem-solving in physics and engineering. It’s all about seeing the same space from a new perspective, which can reveal more about the problem structure or solution. For the exercise, switching from cylindrical to spherical coordinates allowed the equation transformation to reflect a different type of symmetry, broadening the understanding of the space described.