Electromagnetic Theory Lesson 6 (v2) Coordinate Transformations & Scale Factors

Описание: In this lesson we bring together the three coordinate systems (Cartesian, cylindrical, spherical) and show how to convert unit vectors between them. We start with scale factors and a practical mnemonic for remembering them. Then we derive the transformation matrices from the general formula, introduce the transpose trick for inverse transformations, and work through examples — including cases where unit vectors from different coordinate systems appear in the same expression.

Timestamps:
0:00 Intro
0:03 Recap & This Lesson
0:38 Scale Factors & Mnemonic
2:17 Transformation Formula & Cylindrical Matrix
3:53 Spherical Matrix & Transpose Trick
5:20 Worked Examples
7:05 Summary

Topics covered:
Scale factors h_ρ, h_φ, h_θ and how to remember them
Mnemonic: new length coordinate = scale factor of new angle
Right triangle relation: ρ = r sin θ
General unit vector formula: ê_i = (1/h_i)(∂r/∂q_i)
Cartesian ↔ Cylindrical transformation matrix derivation
Cartesian ↔ Spherical transformation matrix derivation
Inverse transformation = transpose (orthogonal matrices)
Converting between cylindrical and spherical via Cartesian
Example: ê_x + ê_y → cylindrical
Example: ê_ρ + ê_θ → Cartesian (mixed coordinate systems)

This video is part of the AcEdumy Electromagnetic Theory series.

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