Spherical coordinates - multivariable calculus

During your summer vacation, you went to Sri Lanka with your friends. It was a blast. Sunbathing, partying, discovering the region. As you were out kayaking, you found a hidden lagoon. But it wasn't marked out on the map, so you had no way of remembering the location. Too bad.

In order to describe positions on spherical surfaces, like the earth's surface, you can use (surprise surprise!) spherical coordinates. It would've been quite cumbersome to work out your position in normal coordinates. Spherical coordinates are tailored for this kind of problem.

Given some radius , set

See the picture below.

Normally, we say and . If so, then there's only one set of coordinates which corresponds to a given position. But we could, as we shall see further on, let and be any real number.

Finally, here's a heads up. When we indicate spherical coordinates, the order of and is important. If you don't get the order right, you might be haunted by Einstein's ghost. Just so you know.

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