When you work in a curvilinear coordinate system, and you need to take gradients or tensor derivatives of scalar, vector, and tensor functions, the gradients of the unit vectors appear in your expressions. The unit vectors have gradients because, in a curvilinear coordinate system, they have orientations that depend on position. I gestured and imagined and guessed these derivatives for a spherical coordinate system by thinking geometrically. I got strange expressions I didn't believe. Then, today, I checked them by painstakingly taking derivatives, and my intuitive derivatives turned out to be exactly correct?
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