Spherical Coordinates are perfect for describing points or regions with symmetry around a central point (the origin), rather than an axis (like cylindrical coordinates).
Scroll through these 3 stages to see how we use distances and angles to define points and solid regions in space.
Any point in Cartesian space (x, y, z) can be mapped to spherical coordinates (ρ, θ, φ), and vice versa. Using the conversion formulas defined in class, find the spherical coordinates for the highlighted point (0, 1/2, √3/2).
Use the sliders to check your answer!
This stage shows why we only need one angle (θ) to sweep a full circle [0, 2π], while the other (φ) only needs a half-circle [0, π] to cover all of 3D space.
If a point lies on the negative side of the y-axis, we don't need φ to swing past π. Instead, we just rotate θ further around the xy-plane to reach that side, and then drop φ down from the positive z-axis.
Use the sliders to find (θ, φ) for the target point (0, -1, 0).
To master triple integrals in spherical coordinates, we first need to get comfortable describing solid regions using inequalities.
Manipulate the sliders to find the boundaries of the highlighted solid. What are the minimum and maximum intervals for each coordinate?