How do you find the area of spherical coordinates?

How do you find the area of spherical coordinates?

On the surface of the sphere, ρ = a, so the coordinates are just the two angles φ and θ. The area element dS is most easily found using the volume element: dV = ρ2 sin φ dρ dφ dθ = dS · dρ = area · thickness so that dividing by the thickness dρ and setting ρ = a, we get (9) dS = a2 sin φ dφ dθ.

What is the area element of a sphere?

An area element on a sphere has constant radius r, and two angles. One is longitude ϕ , which varies from 0 to 2π . The other one is the angle with the vertical. To avoid counting twice, that angle only varies between 0 and π .

What is the Z direction in spherical coordinates?

The spherical coordinate system is defined with respect to the Cartesian system in Figure 4.4. 1. The spherical system uses r, the distance measured from the origin; θ, the angle measured from the +z axis toward the z=0 plane; and ϕ, the angle measured in a plane of constant z, identical to ϕ in the cylindrical system.

How do you find the area of a sphere in spherical coordinates?

i.e. x^2+y^2 = R^2 with z between -R and R. Any region on the sphere has the same area as the corresponding area on the cylinder. The correspondence is via a radial projection out from the z axis. So, for example, the area between latitudes would be 2pi*R^2(cos(phi1)-cos(phi2)).

How do you write a sphere in cylindrical coordinates?

1 Answer

1. x2+y2+z2=R2 .
2. Since x2+y2=r2 in cylindrical coordinates, an equation of the same sphere in cylindrical coordinates can be written as.
3. r2+z2=R2 .

How do you express a sphere in cylindrical coordinates?

To convert a point from spherical coordinates to cylindrical coordinates, use equations r=ρsinφ,θ=θ, and z=ρcosφ.

What is azimuth in spherical coordinates?

In a spherical coordinate system, the azimuth angle refers to the “horizontal angle” between the origin to the point of interest. In Cartesian coordinates, the azimuth angle is the counterclockwise angle from the positive x-axis formed when the point is projected onto the xy-plane.

What is the definition of a spherical coordinate system?

Spherical coordinate system, In geometry, a coordinate system in which any point in three-dimensional space is specified by its angle with respect to a polar axis and angle of rotation with respect to a prime meridian on a sphere of a given radius.

What are the coordinates of a sphere?

In mathematics and physics, spherical polar coordinates (also known as spherical coordinates) form a coordinate system for the three-dimensional real space . Three numbers, two angles and a length specify any point in . The two angles specify the position on the surface of a sphere and the length gives the radius of the sphere.

What are the divergence in spherical coordinates?

The vector (x, y, z) points in the radial direction in spherical coordinates, which we call the direction. Its divergence is 3. A multiplier which will convert its divergence to 0 must therefore have, by the product theorem, a gradient that is multiplied by itself. The function does this very thing, so the 0-divergence function in the direction is

How to convert Cartesian to spherical?

How to transform from Cartesian coordinates to spherical coordinates? We can transform from Cartesian coordinates to spherical coordinates using right triangles, trigonometry, and the Pythagorean theorem . Cartesian coordinates are written in the form (x, y, z), while spherical coordinates have the form (ρ, θ, φ).