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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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Lesson Transcript

Instructor:
*Aaron Miller*

Aaron teaches physics and holds a doctorate in physics.

In this lesson, we introduce two coordinate systems that are useful alternatives to Cartesian coordinates in three dimensions. Both cylindrical and spherical coordinates use angles to specify the locations of points, a feature they share with 2-D polar coordinates.

When you look into the night sky, you can see about five thousand of the more than one hundred billion stars in our Milky Way Galaxy. Humans have charted the stars as long as they have been mapping large-scale features on Earth's surface. So why are maps of Earth's features so widely available, while maps of the locations of nearby stars in our galaxy are hard to come by?

There are likely a few reasons. One big reason is that humans use land and sea maps regularly for navigation, but star maps? Not so much. Another reason is that accurately and meaningfully representing locations of features and relative distances in a three-dimensional space is not easy. The map itself needs to be 3-D if you don't want to lose information by suppressing one of the dimensions. Maps of Earth typically suppress elevation, which is why Earth's surface can be represented on a 2-D map. Let's see how spherical coordinates provide a natural way of representing the locations of stars in our local region of the galaxy.

The idea behind cylindrical and spherical coordinates is to use angles instead of Cartesian coordinates to specify points in three dimensions. Sometimes, employing angles can make mathematical representations of functions simpler. **Polar coordinates** represent points in the coordinate plane, not with the usual Cartesian ordered pair *(x, y)*, but with two different coordinates *(r, phi)* that are functionally related to *(x, y)*. Specifically, for a given point *P*, *r* is the absolute distance from the origin to *P*. The angle *phi* is the angular position of *P*, with angle measured from the positive *x*-axis. Cylindrical and spherical coordinate systems are extensions of 2-D polar coordinates into a 3-D space.

**Cylindrical coordinates** are most similar to 2-D polar coordinates. Let's consider a point *P* that has coordinates *(x, y, z)* in a 3-D Cartesian coordinate system. The same point can be represented in cylindrical coordinates *(r, phi, z)* where *r* and *phi* are the 2-D polar coordinates of *P*'s image in the *xy* plane (*z = 0*), and *z* is exactly the same as *P*'s Cartesian z-coordinate. Here is the relationship between a point's Cartesian and cylindrical coordinates on a graph:

The coordinate transformations to go from Cartesian *x* and *y* coordinates to cylindrical *r* and *phi* coordinates are as follows:

These are the same as the transformation to 2-D polar coordinates.

There are a few features of this transformation to notice. First, the coordinate *r* under this transformation is always a positive number. *r* is interpreted as the smallest distance from *P* to the *z* axis. Also, *phi*, expressed in radians, will always be between *-pi* and *pi*. The *z* coordinate keeps the same value as you transform from one system to the other. The inverse transformation from *(r, theta, z)* to *(x, y, z)* may also be familiar from 2-D polar coordinates as well.

Let's consider the following question as an example of applying the coordinate transformation: what are the Cartesian coordinates *(x, y, z)* of the point P specified by cylindrical coordinates *(2, -pi/6, 1)*?

In moving from cylindrical to Cartesian coordinates, the *z*-coordinate does not change. *z* is conventionally the third value in the ordered triplet, therefore, *z = 1* in both cylindrical and Cartesian coordinates. Now, to find *x* and *y*, we should plug in values *r = 2* and *phi = -pi/6* into the transformation equations.

Therefore, the Cartesian coordinates of this point are *(sqrt(3), -1, 1)*.

Cylindrical coordinates are not the only way to specify a point in a 3-D space using an angle. **Spherical coordinates** are another generalization of 2-D polar coordinates. However, in this coordinate system, there are two angles, *theta* and *phi*.

Let's consider a point *P* that is specified by coordinates *(x, y, z)* in a 3-D Cartesian coordinate system. The same point can be represented in spherical coordinates as *(r, theta, phi,)* where *r*, *theta*, and *phi* are functionally related to *x*, *y*, and *z*, as we will see.

In spherical coordinates, *r* is the distance from the origin to point *P* along the line connecting them. The first angle, *theta*, is often called the **polar angle** because it runs between the 'poles' of the coordinate system, the negative and positive *z*-axes. *Theta* takes the value 0 along the positive *z*-axis and *pi* along the negative *z*-axis. The second angle, *phi*, is called the **azimuth angle**, and it is identical to the angle *phi* in cylindrical coordinates.

The coordinate transformations take a point *P* in Cartesian coordinates to its corresponding spherical coordinates. There are a few features to note in this transformation. The radial coordinate *r* under this transformation is always a positive number that is exactly equal to the Euclidean distance from *P* to the origin.

Also note that the two angles take different ranges of values. Polar angle *theta*, expressed in radians, must always be between 0 and *pi*, but azimuth angle *phi* can point in any direction in the *xy* plane, so it takes values from *-pi* to *pi*.

Based on the geometry, the inverse transformation takes spherical coordinates back to Cartesian coordinates.

Let's consider an example: what are the spherical coordinates *(r, theta, phi)* of the point P specified by Cartesian coordinates *(3, -sqrt(3), -2)*?

In moving from Cartesian to spherical coordinates, we should use these calculations:

Star maps are a common application of spherical coordinates. As observed on Earth, the stars appear to us on the inside of a sphere centered on us. Although early astronomers and philosopher believed the stars were all equidistant from us, we know now that the exact distance to each star we see is different.

In terms of spherical coordinates, the relative positions of the stars in the sky can be specified by two numbers *(theta, phi)*. The actual 3-D distance can be incorporated by given an *r* coordinate for each star. Celestial maps for stargazers tend to use a complementary angle to polar angle *theta*, which is called the **altitude**, to locate a star.

Polar coordinate systems use angles as coordinates of points. Cylindrical and spherical coordinate systems are generalizations of 2-D polar coordinates into three dimensions. **Polar coordinates** represent points in the coordinate plane, not with the usual Cartesian ordered pair (*x*, *y*), but with two different coordinates (*r*, *phi*).

**Cylindrical coordinates** are most similar to 2-D polar coordinates. They use (*r*, *phi*, *z*) where *r* and *phi* are the 2-D polar coordinates of *P*'s image in the *x*-*y* plane and *z* is exactly the same as *P*'s Cartesian *z* coordinate.

In **spherical coordinates**, another angle, the polar angle *theta*, is also defined to specify a point in 3-D. These coordinate systems are used by astronomers and engineers to simplify mathematical models of systems of interest.

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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