Can you help me with both parts? Spherical and cylindrical unit vector decomposition:
Figure 1: Above, left: Spherical coordinates for Earth; Above, right: Cylindrical coordinates for Earth.
Figure 2: Below: Spherical and cylindrical coordinates for Earth.
The spherical coordinates (r, θ, φ) for the point P on the surface of Earth are shown in the figure above. Choose unit vectors (r, θ, φ) at the point P as follows. Let r point radially away from the center of Earth. Let θ point tangent to a circle in the positive φ-direction in the plane formed by the z-axis and the ray from the origin to the point P. Note that φ points in the direction of increasing φ. Choose φ to point in the direction of increasing θ. This unit vector points in the tangent to a circle in the z-y plane centered about the θ-axis.
The unit vectors satisfy the following direct (cross) product relation:
r x θ = φ
In the figure above right, the unit vectors associated with cylindrical coordinates (r, θ, z) are shown.
We have a choice of different coordinates in describing position around Earth, including spherical (r, θ, φ) and cylindrical (r, θ, z).
Part a) Write the cylindrical unit vectors r, θ, and z in terms of the spherical unit vectors r, θ, φ.
Express your answers using some or all of the following: θ, r, φ.
z = rφ
Part b) Write the spherical unit vectors r and φ in terms of the cylindrical unit vectors r, θ, z.
φ = θ