ERROR: Throughout the video, I use dR/du and dR/dv. These should be partial derivatives, not full derivatives. First video on the metric tensor (vector length): https://www.youtube.com/watch?v=C76lWSOTqnc Second video on the metric tensor (arc length in flat space): https://www.youtube.com/watch?v=BbQmTmSzUCI Course notes on Differential Geometry: http://liavas.net/courses/math430/ Why derivative operators are vectors: https://www.youtube.com/watch?v=VHkL5HpL0HY