Misner et al. The Christoffel symbols of the second kind in the definition of Misner et al. The Christoffel symbols of the second kind in the definition of Arfken are given by. Walton ; Moon and Spencer , p. Time derivatives of the unit vectors are. To express partial derivatives with respect to Cartesian axes in terms of partial derivatives of the spherical coordinates,.
The Cartesian partial derivatives in spherical coordinates are therefore. The Helmholtz differential equation is separable in spherical coordinates. Anton, H. Calculus with Analytic Geometry, 2nd ed.
New York: Wiley, Apostol, T. Calculus, 2nd ed. Waltham, MA: Blaisdell, Arfken, G. Orlando, FL: Academic Press, pp. Beyer, W. Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand.
A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the following example, we examine several different problems and discuss how to select the best coordinate system for each one. In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes.
There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem. Which coordinate system is most appropriate for creating a star map, as viewed from Earth see the following figure? Learning Objectives Convert from cylindrical to rectangular coordinates. Convert from rectangular to cylindrical coordinates. Convert from spherical to rectangular coordinates.
Convert from rectangular to spherical coordinates. Cylindrical Coordinates When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. In three dimensions, this same equation describes a half-plane. Spherical Coordinates In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance.
Hint Converting the coordinates first may help to find the location of the point in space more easily. Answer b This set of points forms a half plane. Find the center of gravity of a bowling ball. Determine the velocity of a submarine subjected to an ocean current. Calculate the pressure in a conical water tank.
Find the volume of oil flowing through a pipeline. Determine the amount of leather required to make a football. The origin should be located at the physical center of the ball. Bowling balls normally have a weight block in the center. This looks bad but given that the limits are all constants the integrals here tend to not be too bad. First, we need to take care of the limits.
Therefore, because we are inside a portion of a sphere of radius 2 we must have,. There are two ways to get this. The resulting surface is a cone see the following figure. The traces in planes parallel to the xy -plane are circles. The radius of the circles increases as increases. Describe the surface with cylindrical equation.
This surface is a cylinder with radius. Hint The and components of points on the surface can take any value. In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, location of a point in space is described using two distances and an angle measure In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space.
In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates. In the spherical coordinate system , a point in space Figure is represented by the ordered triple where.
By convention, the origin is represented as in spherical coordinates. Rectangular coordinates and spherical coordinates of a point are related as follows:. If a point has cylindrical coordinates then these equations define the relationship between cylindrical and spherical coordinates. The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry.
Looking at Figure , it is easy to see that Then, looking at the triangle in the xy -plane with as its hypotenuse, we have The derivation of the formula for is similar. Figure also shows that and Solving this last equation for and then substituting from the first equation yields Also, note that, as before, we must be careful when using the formula to choose the correct value of.
Let be a constant, and consider surfaces of the form Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form are half-planes, as before.
Last, consider surfaces of the form The points on these surfaces are at a fixed angle from the z -axis and form a half-cone Figure. Converting from Spherical Coordinates Plot the point with spherical coordinates and express its location in both rectangular and cylindrical coordinates. Use the equations in Figure to translate between spherical and cylindrical coordinates Figure :. The point with spherical coordinates has rectangular coordinates.
Finding the values in cylindrical coordinates is equally straightforward:. Thus, cylindrical coordinates for the point are. Plot the point with spherical coordinates and describe its location in both rectangular and cylindrical coordinates.
Cartesian: cylindrical:. Converting the coordinates first may help to find the location of the point in space more easily. Convert the rectangular coordinates to both spherical and cylindrical coordinates. Because then the correct choice for is. There are actually two ways to identify We can use the equation A more simple approach, however, is to use equation We know that and so. To find the cylindrical coordinates for the point, we need only find. The cylindrical coordinates for the point are.
Equation describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring rad with the positive z -axis. These points form a half-cone Figure. Because there is only one value for that is measured from the positive z -axis, we do not get the full cone with two pieces.
The equation describes a cone. To find the equation in rectangular coordinates, use equation. This is the equation of a cone centered on the z -axis. Equation describes the set of all points units away from the origin—a sphere with radius Figure. Equation describes a sphere with radius To identify this surface, convert the equation from spherical to rectangular coordinates, using equations and The equation describes a sphere centered at point with radius Describe the surfaces defined by the following equations.
This is the set of all points units from the origin. This set forms a sphere with radius b. This set of points forms a half plane. The angle between the half plane and the positive x -axis is c.
Let be a point on this surface. The position vector of this point forms an angle of with the positive z -axis, which means that points closer to the origin are closer to the axis.
These points form a half-cone. Think about what each component represents and what it means to hold that component constant. A sphere that has Cartesian equation has the simple equation in spherical coordinates. Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth.
We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees. Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive z -axis.
The prime meridian represents the trace of the surface as it intersects the xz -plane. The equator is the trace of the sphere intersecting the xy -plane. The latitude of Columbus, Ohio, is N and the longitude is W, which means that Columbus is north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is In the same way, measuring from the prime meridian, Columbus lies to the west.
Express the location of Columbus in spherical coordinates. The radius of Earth is mi, so The intersection of the prime meridian and the equator lies on the positive x -axis.
Movement to the west is then described with negative angle measures, which shows that Because Columbus lies north of the equator, it lies south of the North Pole, so In spherical coordinates, Columbus lies at point.
Because Sydney lies south of the equator, we need to add to find the angle measured from the positive z -axis.
Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations.
In the following example, we examine several different problems and discuss how to select the best coordinate system for each one. In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem.
Note : There is not enough information to set up or solve these problems; we simply select the coordinate system Figure.
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