ezLecture: Working and Deriving in Different Coordinate Systems
Worksheet
Learning Goals:
1. Define the displacement vector in non-Cartesian coordinate systems, such as polar
coordinates, spherical polar coordinates, and cylindrical polar coordinates.
2. Calculate time derivatives in these non-Cartesian coordinate systems.
3. Calculate spatial integrals in these non-Cartesian coordinate systems.
1. Define the displacement vector in non-Cartesian coordinate systems.
For each system below, draw a graph with the point P and label the corresponding coordinates. Then,
write down the displacement vector in that coordinate system in both the parenthetical notation and in
terms of the unit vectors.
Cartesian
𝑟⃗ = ( ,
,
𝑟⃗ =
)
Polar
𝑟⃗ = (
𝑟⃗ =
,
)
Cylindrical Polar
𝑟⃗ = (
, , )
𝑟⃗ =
Spherical Polar
𝑟⃗ = (
, , )
𝑟⃗ =
Give the mathematical expression for each of the coordinates or unit vectors below in terms of the
Cartesian coordinates or unit vectors.
Cylindrical polar 𝜑 =
Spherical polar 𝑟 =
Cylindrical polar 𝜌 =
Polar 𝑟̂ =
Spherical polar 𝜃̂ =
Polar 𝜑̂ =
Given the 2D and 3D graphs below, write the displacement vector in both the parenthetical notation
and in terms of the unit vectors.
Cartesian
𝑟⃗ = (
,
𝑟⃗ =
,
)
Polar
𝑟⃗ = (
𝑟⃗ =
,
)
Cylindrical Polar
𝑟⃗ = (
, , )
𝑟⃗ =
Spherical Polar
𝑟⃗ = (
, , )
𝑟⃗ =
Given the following parametric equations for x, y, and z; write down the displacement vector in unit
vector notation for each of the following coordinate systems. For polar coordinates, ignore the z term.
𝑥 = 𝑡 sin(2𝑡)
𝑦 = 𝑡 cos(2𝑡)
𝑧=𝑡
Cartesian
𝑟⃗ =
Polar
𝑟⃗ =
Cylindrical Polar
𝑟⃗ =
Spherical Polar
𝑟⃗ =
For more information see Classical Mechanics by Taylor: polar coordinates (pg. 26, 34), polar cylindrical
coordinates (pg. 34, 40), and polar spherical coordinates (pg. 135).
2. Calculate time derivatives in non-Cartesian coordinate systems.
Write down the displacement vector and calculate the velocity and acceleration vectors for the
following non-Cartesian coordinate systems by taking the time derivatives.
Polar
𝑟⃗ =
𝑣⃗ =
Cylindrical Polar
𝑟⃗ =
𝑣⃗ =
Spherical Polar
𝑟⃗ =
𝑣⃗ =
𝑎⃗ =
𝑎⃗ =
𝑎⃗ =
Using the polar coordinate system and the following parametric equations, give the velocity and
acceleration vectors in terms of 𝑡 and the unit vectors 𝑟̂ and 𝜑̂.
𝑟 = 𝑡 3 sin(2𝑡)
𝜑 = 𝑡 2 (𝑡 − 1)
𝑣⃗ =
𝑎⃗ =
3. Calculate spatial integrals in non-Cartesian coordinate systems.
Use a spatial integral in polar coordinates to prove that the area of a circle with radius 𝑅 is 𝜋𝑅 2.
Use a spatial integral in cylindrical polar coordinates to prove that the volume of a cylinder with radius 𝑅
and length 𝐿 is 𝜋𝑅 2 𝐿.
Use a spatial integral in spherical polar coordinates to prove that the volume of a sphere with radius 𝑅 is
4
𝜋𝑅 3 .
3