Problem 36 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The shape of the given set in spherical coordinates is an elliptic cylinder with an ellipse in the xy-plane having a semi-major and semi-minor axes of 2, and extending infinitely along the z-axis.

Step 1: Recall the spherical coordinate system conversion

To have a better understanding of what the equation represents, recall the conversion between spherical and Cartesian coordinates: $$ x = \rho \sin\varphi \cos\theta $$ $$ y = \rho \sin\varphi \sin\theta $$ $$ z = \rho \cos\varphi $$

Step 2: Plug in the given equation

Now, substitute the given equation, \(\rho=2 \csc \varphi\) into the conversion equations. Keep in mind that \(\csc\varphi = \frac{1}{\sin\varphi}\), so the given equation can also be written as \(\rho=2\frac{1}{\sin\varphi}\): $$ x = (2\frac{1}{\sin\varphi}) \sin\varphi \cos\theta = 2\cos\theta $$ $$ y = (2\frac{1}{\sin\varphi}) \sin\varphi \sin\theta = 2\sin\theta $$ $$ z = (2\frac{1}{\sin\varphi}) \cos\varphi = 2\cot\varphi $$

Step 3: Identify the shape

The shape of the set can be determined from the equation of \(x\), \(y\), and \(z\). In this case, we see that: $$ x^2 = 4\cos^2\theta $$ $$ y^2 = 4\sin^2\theta $$ Combining the two equations: $$ \frac{x^2}{4} + \frac{y^2}{4} = \cos^2\theta + \sin^2\theta = 1 $$ Which represents an ellipse in the \(xy\)-plane with semi-major axis 2 and semi-minor axis 2. Considering the equation for \(z\): $$ z = 2\cot\varphi $$ As \(\varphi\) goes from \(0\) to \(\pi\), \(z\) goes from \(+\infty\) to \(-\infty\). This means the shape extends infinitely in both positive and negative \(z\)-directions.

Step 4: Sketch the shape

Based on the identified shape, it is an elliptic cylinder with the ellipse in the \(xy\)-plane and extending infinitely along the \(z\)-axis. To sketch this shape, draw an ellipse in the \(xy\)-plane with the center at the origin, whose semi-major and semi-minor axes are both 2. Then, extend the ellipse infinitely along the \(z\)-axis in both positive and negative directions.