Question Video: Expressing Cartesian Functions in Cylindrical and Spherical Coordinates

Express the equation 𝑥² + 𝑦² = 2𝑦 in cylindrical and spherical coordinates.

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Video Transcript

Express the equation 𝑥 squared plus 𝑦 squared equals two 𝑦 in cylindrical and spherical coordinates.

Starting with cylindrical coordinates, the Cartesian coordinates 𝑥, 𝑦, and 𝑧 are given in cylindricals by 𝑥 equals 𝜌 cos 𝜃, 𝑦 equals 𝜌 sin 𝜃, and 𝑧 is unchanged. So, 𝑧 is just the distance of the point along the positive 𝑧-axis, 𝜌 is the horizontal distance of the point from the 𝑧-axis, and 𝜃 is the horizontal argument that the point makes with the positive 𝑥-axis. With this in mind, we can replace any 𝑥, 𝑦, and 𝑧 in our equation with these expressions.

So, our equation in cylindrical coordinates becomes 𝜌 squared cos squared 𝜃 plus 𝜌 squared sin squared 𝜃 equals two 𝜌 sin 𝜃. Now, we can use the trigonometric identity cos squared 𝜃 plus sin squared 𝜃 is identically equal to one to reexpress the left-hand side. Taking out a common factor of 𝜌 squared, on the left-hand side, we have 𝜌 squared times cos square 𝜃 plus sin squared 𝜃. And using our identity, this whole term is identically equal to one. So, this gives us 𝜌 squared equals two 𝜌 sin 𝜃. Dividing both sides by 𝜌 gives us the equation in cylindrical coordinates 𝜌 equals two sin 𝜃.

Moving on to spherical coordinates, the Cartesian coordinates 𝑥, 𝑦, and 𝑧 are given in sphericals by 𝑥 equals 𝑟 cos 𝜃 sin 𝜙, 𝑦 equals 𝑟 sin 𝜃 sin 𝜙, and 𝑧 equals 𝑟 cos 𝜙, where 𝑟 is the distance of the point from the origin, 𝜙 is the argument that the point makes with the positive 𝑧-axis, and 𝜃 is the horizontal argument the point makes with the positive 𝑥-axis. As before, we can replace these expressions into our equation. So, this gives us 𝑟 squared cos squared 𝜃 sin squared 𝜙 plus 𝑟 squared sin squared 𝜃 sin squared 𝜙 equals two 𝑟 sin 𝜃 sin 𝜙.

On the left-hand side, we can factor out a common factor of 𝑟 squared sin squared 𝜙. And inside the parentheses, we once again have cos squared 𝜃 plus sin squared 𝜃, which is identically equal to one. This leaves us with 𝑟 squared sin squared 𝜙 equals two 𝑟 sin 𝜃 sin 𝜙. And dividing both sides by the common factor of 𝑟 sin 𝜙 leaves us with the equation in spherical coordinates 𝑟 sin 𝜙 equals two sin 𝜃.

Notice that the right-hand side of the equation in both sphericals and cylindricals is the same. This implies that the horizontal distance from the 𝑧-axis in cylindricals, 𝜌, is given by the distance from the origin, 𝑟, times sin 𝜙, the argument with the positive 𝑧-axis, which indeed it is. So, this provides a handy way for you to check your answer.

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