Abstract:

This lecture, presented by Professor Karl-Heinz Lotze as part of the "Mathematical Methods of Physics" course at the University of Jena, provides a systematic introduction to spherical coordinates. The presentation begins by deriving spherical coordinates (r, θ, φ) from the more familiar cylindrical coordinates, establishing the transformation equations from the Cartesian system. Professor Lotze explains the physical and geometric meaning of each coordinate—radial distance (r), polar angle or colatitude (θ), and azimuthal angle (φ)—and defines their respective valid intervals. The core of the lecture is dedicated to deriving the fundamental components of this curvilinear orthogonal system: the tangent vectors along the coordinate lines, the line element (ds²), the orthogonal area elements (dF) corresponding to surfaces of constant r, θ, and φ, and finally, the volume element (dV). The practical utility of this framework is demonstrated by integrating the derived volume element to calculate the volume of a sphere, confirming the classic formula V = (4/3)πR³.

Deriving and Applying Spherical Coordinates in Physics

• 0:00 Introduction to Spherical Coordinates: The lecture introduces spherical coordinates as one of the most common curvilinear orthogonal systems, outlining the goal to derive them and calculate their line, area, and volume elements.
• 1:23 Derivation from Cylindrical Coordinates: The lecture builds upon previous knowledge by starting with cylindrical coordinates (ρ, φ, z) and showing how they are transformed into spherical coordinates (r, θ, φ). The key is expressing the cylindrical radius (ρ) and height (z) using the spherical radius (r) and polar angle (θ).
• 5:13 Defining the Coordinates and Their Ranges: The lecture provides clear definitions and analogies for the coordinates: r is the radial distance from the origin, θ is the polar angle measured from the z-axis (Poldistanz), and φ is the azimuthal angle. It also gives the standard intervals: r ∈ [0, ∞), θ ∈ [0, π], and φ ∈ [0, 2π).
• 14:17 Calculating Tangent Vectors: The process of finding the tangent vectors for the coordinate lines begins. This is done by taking the partial derivatives of the position vector with respect to each of the three spherical coordinates (r, θ, and φ).
• 21:54 Scale Factors and Orthogonality: The derivations show that the tangent vectors are mutually orthogonal. The lengths of these infinitesimal vectors reveal the crucial scale factors for each coordinate direction: 1 for the r-direction, r for the θ-direction, and r sin(θ) for the φ-direction.
• 40:23 The Line Element (ds²): The square of the infinitesimal distance between two points is derived. The resulting formula, ds² = dr² + r²dθ² + r²sin²(θ)dφ², lacks any cross-terms (like drdθ), which mathematically confirms that the coordinate system is orthogonal.
• 43:31 Deriving Area Elements: The lecture explains how to find the three distinct, mutually orthogonal infinitesimal area elements (dF) by taking the cross product of pairs of tangent vectors.
• 46:03 Physical Meaning of Area Elements: Each area element is given a clear geometric interpretation:
  • dF₁ (constant r): An area element on the surface of a sphere. Integrating this yields the sphere's surface area, 4πR².
  • dF₂ (constant φ): An area element of a meridian plane (a semi-circle from pole to pole).
  • dF₃ (constant θ): An area element on the mantle (side surface) of a cone.
• 56:45 The Volume Element (dV): The infinitesimal volume element is calculated using the scalar triple product of the three tangent vectors, resulting in the fundamental formula dV = r² sin(θ) dr dθ dφ.
• 1:00:42 Key Application: Calculating the Volume of a Sphere: As a final, practical demonstration, the lecture uses the derived volume element (dV) and integrates it over the full range of r, θ, and φ to derive the well-known formula for the volume of a sphere, V = (4/3)πR³, showcasing the power of using a coordinate system adapted to a problem's symmetry.