All the examples of paramaterized surfaces in Lecture 1 were examples of this. For instance, if M = S², the 2-sphere, and if x = (x¹, x²) denotes the chart for polar coordinates on the sphere, then x^-1 is given by

y₁ = sin(x¹)cos(x²)
y₂ = sin(x¹)sin(x²)
y₃ = cos(x¹)

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