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Integration in Spherical Coordinates
Integration in Spherical Coordinates

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Spherical Coordinates

Spherical coordinates are a set of three numbers that form an ordered triplet and are used to describe a point in the spherical coordinate system. Spherical coordinates use the radial distance, the polar angle, and the azimuthal angle of the orthogonal projection to locate a point in three-dimensional space.

Spherical coordinates are used in Geography to communicate various locations or points on the Earth. Spherical coordinates usually use radians rather than degrees to depict angles related to the position of a point. In this article, we will learn more about spherical coordinates, the spherical coordinate system, various conversions, and associated examples.

1. What are Spherical Coordinates?
2. What is a Spherical Coordinate System?
3. Spherical Coordinates Conversions
4. Jacobian For Spherical Coordinates
5. Applications of Spherical Coordinate System
6. FAQs on Spherical Coordinates

What are Spherical Coordinates?

Spherical coordinates are ordered triplets used to describe the location of a point in the spherical coordinate system. In each spherical coordinate triplet, one number represents the distance while the other two denote angles. A spherical coordinate system is represented as follows:

Here, ρ represents the distance between point P and the origin. The spherical coordinates of the origin, O, are (0, 0, 0). The value of ρ should be greater than or equal to 0, i.e., ρ ≥ 0.

θ is used to describe the location of P. Let Q be the projection of point P on the xy plane. Then the angle between the line segment drawn to point Q from the origin and the positive x-axis is represented by θ. No restrictions are imposed on θ.

φ is the angle between the line segment from the origin to P and the positive z-axis. This is also known as the azimuthal angle and the restriction is 0≤ φ ≤ π.

Thus, the coordinates of P are given as (ρ,θ,φ).

What is a Spherical Coordinate System?

The spherical coordinate system is a three-dimensional system that is used to describe a sphere or a spheroid. By using a spherical coordinate system, it becomes much easier to work with points on a spherical surface. For example, the cartesian equation of a sphere is given by x2 + y2 + z2 = c2. However, in the spherical coordinate system, this equation will simply be represented as ρ = c. The location of a point on the Earth can be described by spherical coordinates as the Earth is analogous to a spherical coordinate system. The following conventions are used to locate a point on Earth using spherical coordinates.

The coordinate ρ will correspond to the altitude. It indicates how far a point is from the origin. On Earth, it is used to measure the distance below or above sea level.

θ will correspond to the longitude and is used to measure the angular distance from the horizontal axis.

φ will represent the latitude and measure the angular distance from the North Pole.

Spherical Coordinates Conversions

There are many coordinate systems that exist in three dimensions. One of them is the spherical coordinate system. Thus, there exist different conversion formulas that can be used to represent the coordinates of a point in different systems.

Spherical Coordinates to Cylindrical Coordinates

To convert spherical coordinates (ρ,θ,φ) to cylindrical coordinates (r,θ,z), the derivation is given as follows:

Given above is a right-angled triangle. Using trigonometry, z and r can be expressed as follows:

z = ρcosφ

r = ρsinφ

As θ is the same in both coordinate systems we can express the cylindrical coordinates in the form of spherical coordinates as follows:

r = ρsinφ

θ = θ

z = ρcosφ

Cylinderical Coordinates to Spherical Coordinates

In order to convert cylindrical coordinates to spherical coordinates, the following equations are used.

\(\rho =\sqrt{r^{2}+z^{2}}\)

θ = θ

φ = \(cos^{-1}\left ( \frac{z}{\sqrt{r^{2}+z^{2}}} \right )\)

Spherical Coordinates to Cartesian Coordinates

Cartesian coordinates can also be referred to as rectangular coordinates. The derivation is given as follows:

The figure given above represents a point in a cartesian coordinate system. Here, (x, y, z) shows the cartesian coordinates of the point, and (r,θ,z) shows its corresponding cylindrical coordinates.

Using trigonometry, we get

x = rcosθ

y = rsinθ

z = z

However, we already know that

z = ρcosφ

r = ρsinφ

On substituting we get,

x = ρsinφcosθ

y = ρsinφsinθ

z = ρcosφ

Thus, these equations convert spherical coordinates to cartesian coordinates.

Cartesian Coordinates to Spherical Coordinates

The equations given below are used to convert rectangular coordinates to spherical coordinates:

ρ2 = x2 + y2 + z2

tanθ = y / x

φ = \(cos^{-1}\left ( \frac{z}{\sqrt{x^{2}+y^{2} +z^{2}}} \right )\)

Jacobian For Spherical Coordinates

A Jacobian matrix can be defined as a matrix that consists of all the first-order partial derivatives of a vector function with several variables. The Jacobian matrix for spherical coordinates transformation to cartesian coordinates is given as follows:

x = ρsinφcosθ

y = ρsinφsinθ

z = ρcosφ

\(J(\rho ,\theta ,\phi ) = \begin{bmatrix} \frac{\partial x}{\partial\rho } & \frac{\partial x}{\partial\theta } &\frac{\partial x}{\partial\phi } \\ \frac{\partial y}{\partial\rho } & \frac{\partial y}{\partial\theta } &\frac{\partial y}{\partial\phi } \\ \frac{\partial z}{\partial\rho } & \frac{\partial z}{\partial\theta } &\frac{\partial z}{\partial\phi } \end{bmatrix}\)

\(\frac{\partial (x, y, z)}{\partial (\rho, \theta, \phi)}\) = \(J(\rho ,\theta ,\phi ) = \begin{bmatrix} cos\theta sin\phi & -\rho sin\theta sin\phi &\rho cos\theta cos\phi \\ sin\theta sin\phi & \rho cos\theta sin\phi &\rho sin\theta cos\phi \\ cos\phi & 0 &-\rho sin\phi \end{bmatrix}\)

The determinant of this Jacobian matrix is ρ2sinφ.

Spherical Coordinates Integral

The volume element helps to integrate a function in different coordinate systems. Now if the volume element needs to be transformed using spherical coordinates then the algorithm is given as follows:

The volume element is represented by dV = dx dy dz.

The transformation formula for the volume element is given as

dV = \(|\frac{\partial (x, y, z)}{\partial (\rho, \theta, \phi)}|\) \(d\overline{V}\)

Using the result from the Jacobian matrix for spherical coordinates this equation can be expressed as:

dV = ρ2sinφ dρdθdφ

This result can be used to transform triple integrals from one coordinate system to another. Let f(x, y, z) be the function whose volume needs to be determined in a spherical coordinate system. Then the spherical coordinate integral is given below:

\(\int \int \int f(x,y,z)dxdydz\) =

\(\int \int \int f(\rho sin\phi cos\theta,\rho sin\phi sin\theta, \rho cos\phi)\rho^{2}sin\phi\: d\rho d\theta d\phi\)

Applications of Spherical Coordinate System

Spherical coordinates are very useful in analyzing systems that demonstrate some form of symmetry. Other applications of the spherical coordinate system are listed below:

  • The coordinates can be used to find the volume integral within a sphere.
  • Using the spherical coordinate system three-dimensional modeling of the output of a loudspeaker can be done.
  • A simulation of the Earth’s geographical conditions can be modeled using spherical coordinates.

Related Articles:

Important Notes on Spherical Coordinates

  • Spherical coordinates are ordered triplets in the spherical coordinate system and are used to describe the location of a point.
  • The spherical coordinates are represented as (ρ,θ,φ).
  • The determinant of a Jacobian matrix for spherical coordinates is equal to ρ2sinφ.
  • When a volume element has to be transformed from a cartesian coordinate system to a spherical coordinate system then the formula is given as \(\int \int \int f(x,y,z)dxdydz = \int \int \int f(\rho sin\phi cos\theta,\rho sin\phi sin\theta, \rho cos\phi)\rho^{2}sin\phi\: d\rho d\theta d\phi\).

Examples on Spherical Coordinates

  1. Example 1: Express the spherical coordinates (8, π / 3, π / 6) in rectangular coordinates.

    Solution: To perform the conversion from spherical coordinates to rectangular coordinates the equations used are as follows:

    x = ρsinφcosθ

    = 8 sin (π / 6) cos (π / 3)

    x = 2

    y = ρsinφsinθ

    = 8 sin (π / 6) sin (π / 3)

    y = \(2\sqrt{3}\)

    z = ρcosφ

    = 8 cos (π / 6)

    z = \(4\sqrt{3}\)

    Answer: The spherical coordinates (8, π / 3, π / 6) can be converted to cartesian coordinates as (2, \(2\sqrt{3}\), \(4\sqrt{3}\))

  2. Example 2: Convert the spherical coordinates (2, -5π / 6, π / 6) to cylindrical coordinates.

    Solution: The equations to convert spherical coordinates to cylindrical coordinates are as follows:

    r = ρsinφ

    = 2 sin (π / 6)

    = 1

    θ = -5π / 6

    z = ρcosφ

    = 2 cos (π / 6)

    = \(\sqrt{3}\)

    Answer: The spherical coordinates (2, -5π / 6, π / 6) can be converted to the cylindrical coordinates (1, -5π / 6, \(\sqrt{3}\))

  3. Example 3: Evaluate the integral \(\int \int \int 16zdV\) in the upper half of the sphere given by the equation x2 + y2 + z2 = 1. The constraints are given as follows:

    0 ≤ ρ ≤ 1

    0 ≤ θ ≤ 2π

    0 ≤ φ ≤ π / 2

    Solution: The integral can be expressed as

    \(\int \int \int 16zdV\) = \(\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{0}^{1}\left (16\rho cos \phi \right )\, \rho^{2}sin\phi \: d\rho d\theta d\phi\)

    (Using trigonometric identities)

    = \(\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}\int_{0}^{1} 8 \rho^{3}sin(2\phi) d\rho d\theta d\phi\)

    = \(\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi} 2sin(2\phi) d\theta d\phi\)

    = \(\int_{0}^{\frac{\pi}{2}} 4\pi sin(2\phi) d\phi\)

    = \([-2\pi cos(2\phi)]_{0}^{\frac{\pi}{2}}\)

    = 4\(\pi\)

    Answer: \(\int \int \int 16zdV\) = 4\(\pi\)

Practice Questions on Spherical Coordinates

FAQs on Spherical Coordinates

What are Spherical Coordinates in a Spherical Coordinate System?

The location of any point in a spherical coordinate system can be described by a set of ordered triplets known as spherical coordinates. These are represented as (ρ,θ,φ).

What is the Definition of a Spherical Coordinate System?

A spherical coordinate system is a three-dimensional curvilinear coordinate system that can be used to describe a point using the radial distance, the polar angle, and the azimuthal angle.

How to Convert Spherical Coordinates for Triple Integrals?

To convert spherical coordinates for triple integrals the Jacobian matrix is used. The transformation is given as \(\int \int \int f(x,y,z)dxdydz\)

= \(\int \int \int f(\rho sin\phi cos\theta,\rho sin\phi sin\theta, \rho cos\phi)\rho^{2}sin\phi\: d\rho d\theta d\phi\).

How to Convert a Vector from Cartesian to Spherical Coordinates?

To convert a vector from cartesian to spherical coordinates the equations are given as follows:

  • ρ2 = x2 + y2 + z2
  • tanθ = y / x
  • φ = \(cos^{-1}\left ( \frac{z}{\sqrt{x^{2}+y^{2} +z^{2}}} \right )\)

What is the Determinant of the Jacobian matrix of Spherical Coordinates?

Using the spherical coordinate transformation the cartesian coordinates are specified as x = ρsinφcosθ, y = ρsinφsinθ, and z = ρcosφ. Using these values, the determinant of the Jacobian matrix is given by ρ2sinφ.

What is the Volume Element in Spherical Coordinates?

The volume element in spherical coordinates can be given by dV = \(|\frac{\partial (x, y, z)}{\partial (\rho, \theta, \phi)}|\) \(d\overline{V}\). On solving this equation, the formula for the volume element can be given as dV = ρ2sinφ dρdθdφ.

What are the Applications of a Spherical Coordinate System?

A spherical coordinate system is very useful in analyzing many natural phenomena related to the Earth such as weather patterns, potential energy flow, etc. Additionally, they are also very useful in evaluating systems that have some form of symmetry.

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