Why does phi go from 0 to pi?

You only need to integrate phi from 0 to pi to sweep out the full volume of the sphere.

What does phi represent in spherical coordinates?

The coordinates used in spherical coordinates are rho, theta, and phi. Rho is the distance from the origin to the point. Theta is the same as the angle used in polar coordinates. Phi is the angle between the z-axis and the line connecting the origin and the point.

What is the first Octant in spherical coordinates?

sphere of x2 + y2 + z2 = 9 in the first octant. The change to spherical coordinates in the function results in f = exp( √ ρ2) = exp(ρ).

What are the coordinates of spherical coordinate system?

Spherical coordinates (r, θ, φ) as commonly used in physics (ISO 80000-2:2019 convention): radial distance r (distance to origin), polar angle θ (theta) (angle with respect to polar axis), and azimuthal angle φ (phi) (angle of rotation from the initial meridian plane).

What is PHI for half a sphere?

Using this, I inferred that since the y-axis splits the sphere in half, the value of φ is π, and since the hemisphere forms a shadow of a full circle on the x-z axis the value for θ is 2π.

What is azimuth angle 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.

Why do we need to transfer Cartesian coordinates to spherical coordinates?

Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation x2+y2+z2=c2 has the simple equation ρ=c in spherical coordinates.

What are the bounds of the first Octant?

z3√x2 + y2 + z2dV , where D is the region in the first octant which is bounded by x = 0, y = 0, z = √x2 + y2, and z = √1 − (x2 + y2). Express this integral as an iterated integral in both cylindrical and spherical coordinates.

How do you integrate the surface of a sphere?

To do the integration, we use spherical coordinates ρ, φ, θ. On the surface of the sphere, ρ = a, so the coordinates are just the two angles φ and θ.

Why do we prefer spherical polar coordinates?

The two angles specify the position on the surface of a sphere and the length gives the radius of the sphere. Spherical polar coordinates are useful in cases where there is (approximate) spherical symmetry, in interactions or in boundary conditions (or in both).