Introduction to spherical coordinates
We look at spherical coordinates, describing regions with spherical coordinates, and converting dV into the appropriate spherical coordinates expression. Imagine looking at a globe. The surface of the Earth, being roughly spherical, is mapped out using a grid of latitude and longitude lines. Latitude measures how far north or south we are from the equator, while longitude measures how far east or west we are from a prime meridian (GMT, Greenwich Mean Time in London). This system of latitude and longitude is a real-world example of a spherical coordinate system, which helps us pinpoint any location on the surface of our planet.
Just as we use latitude and longitude to locate points on the Earth's surface, we use spherical coordinates to specify points on a sphere in ℝ^3. We use the familiar angle 𝜃 to describe a point's position in a plane. This gives us a sense of rotation around the 𝑧-axis and it is analogous to Earth longitude. As always, we measure 𝜃 off of the positive 𝑥-axis (GMT).
The counterpart to latitude in spherical coordinates is the angle 𝜙, but instead of measuring from the equator, we measure 𝜙 from the positive 𝑧-axis. Imagine a line going straight out from the top of the globe (the North Pole) to space; this is our starting point for measuring 𝜙: 𝜙 = 0 is the North Pole. If we rotate down to the 𝑥𝑦-plane (the equator), we arrive at 𝜙=𝜋/2. Lastly all the way down at the South Pole we have 𝜙 = 𝜋.
Finally, while latitude and longitude define a position on a surface, spherical coordinates also need to specify how far out from the center of the sphere (or in mathematical terms, the origin) our point is. This distance is denoted as 𝜌, representing the radius from the origin to our point in space.
In summary, spherical coordinates (𝜌,𝜃,𝜙) in mathematics are akin to a three-dimensional extension of the Earth's system of latitude and longitude:
Distance (𝜌): represents the distance from the origin to a point in space, always non-negative.
Angle 𝜃: the angle a point makes with the positive 𝑥-axis in the 𝑥𝑦-plane.
Angle 𝜙: represents the angle from the positive 𝑧-axis to our point. It ranges from 0 (North Pole) to 𝜋 (South Pole).
When we introduce polar coordinates, we needed to know how to convert "𝑑𝑥𝑑𝑦" into polar differentials. The result was "𝑟𝑑𝑟𝑑𝜃." Now we need to figure out the conversion for spherical coordinates.
Consider a small segment of a sphere. Its volume element is approximated as:
Volume ≈ Length×Width×Height = (𝜌Δ𝜙) × (𝜌sin(𝜙)Δ𝜃) × Δ𝜌 = 𝜌^2sin(𝜙) Δ𝜙Δ𝜃Δ𝜌.
Thus, in spherical coordinates, the differential volume element becomes 𝑑𝑉 = 𝜌^2sin(𝜙)𝑑𝜌𝑑𝜃𝑑𝜙. So converting "𝑑𝑥𝑑𝑦𝑑𝑧" to spherical will need "𝜌^2sin(𝜙)𝑑𝜌𝑑𝜃𝑑𝜙."
Multivariable Calculus Unit 5 Lecture 4.
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