Abstract:

This German-language lecture by Prof. Karl-Heinz Lotze serves as an introduction to curvilinear orthogonal coordinates, focusing specifically on plane polar coordinates (r, φ). The video begins by establishing the fundamental conversion formulas between Cartesian (x, y) and polar coordinates. It then introduces the concepts of coordinate lines (circles for constant r, rays for constant φ), which explains the "curvilinear" nature of the system. The lecture derives the tangent vectors and position-dependent basis vectors (e_r, e_φ), demonstrating that they are mutually perpendicular, hence "orthogonal." Key mathematical tools, the line element (ds) and the area element (dA), are derived formally. To illustrate the practical advantage of using a coordinate system adapted to a problem's geometry, the lecture calculates the circumference and area of a circle using both polar and Cartesian coordinates, highlighting the significant simplification offered by the former. The video concludes with a more intuitive geometric derivation of the line and area elements.

A Physics Lecture on Plane Polar Coordinates

• 0:00 Introduction to Curvilinear Coordinates: The lecture is introduced as an "intermezzo" in a vector analysis series, dedicated to curvilinear orthogonal coordinates. This segment focuses on the most common example: plane polar coordinates.
• 3:01 Defining Polar Coordinates (r, φ): A point P in a 2D plane is defined not by its Cartesian (x, y) coordinates, but by its distance from the origin, r, and the angle φ (azimuthal angle) measured counter-clockwise from the positive x-axis.
• 5:47 Conversion between Coordinate Systems: The fundamental formulas for converting between Cartesian and polar coordinates are presented:
  • x = r cos(φ) and y = r sin(φ)
  • r = √(x² + y²) and tan(φ) = y/x
• 9:05 Coordinate Lines and Orthogonality: The concept of "coordinate lines" is explained. In this system, lines of constant r are circles, and lines of constant φ are rays from the origin. Since these lines intersect at right angles, the system is described as "orthogonal."
• 18:25 Position-Dependent Basis Vectors: The unit basis vectors for the radial (e_r) and azimuthal (e_φ) directions are derived. A critical point is that, unlike the fixed i and j vectors in a Cartesian system, these basis vectors change their direction depending on the angle φ.
• 29:19 Derivatives of Basis Vectors: The lecture calculates how the basis vectors change with the angle φ, a concept crucial for calculating velocity and acceleration in physics. The results are d(e_r)/dφ = e_φ and d(e_φ)/dφ = -e_r.
• 32:36 The Line Element (ds): The infinitesimal distance or "line element" ds is derived. Its square is given by ds² = dr² + r²dφ². The lack of a mixed dr dφ term is a key feature of orthogonal systems.
• 37:21 Application: Arc Length and Circumference: The power of polar coordinates is demonstrated by calculating the arc length of a circle. The calculation is trivial compared to the Cartesian approach, easily yielding the circumference C = 2πR.
• 44:24 Cartesian vs. Polar: A Comparison: To emphasize the benefit of choosing the right coordinates, the lecturer sets up the circle circumference calculation using Cartesian coordinates. This results in a complex integral that can only be solved by a substitution that is, in essence, a conversion to polar coordinates, proving the inefficiency of the Cartesian approach for this geometry.
• 47:57 The Area Element (dA): The infinitesimal area element is derived and found to be dA = r dr dφ. This element represents the area of a small patch bounded by changes in r and φ.
• 52:07 Calculating the Area of a Circle: Using the area element, the total area of a circle is calculated with a simple double integral, resulting in the familiar formula A = πR².
• 55:13 Intuitive Geometric Approach: As a final note, the lecturer shows how the line and area elements can also be understood intuitively using the Pythagorean theorem on an infinitesimal triangle and the formula for the area of a circular sector, respectively.