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## Main Question or Discussion Point

How can the derivative of a basis vector at a point be the linear combination of tangent vectors at that point?

For example, if you take a sphere, then the derivative of the polar basis vector with respect to the polar coordinate is in the radial direction. How can something in the radial direction be written as a linear combination of vectors in a plane perpendicular to the radial direction?

[tex]\partial_i e_j=\Gamma^{k}_{ij} e_k [/tex]

Or is this a misunderstanding of the Christoffel symbols? The above equation seems to make sense in spherical coordinates in 3-dimensional space, but not in spherical coordinates on a 2-dimensional sphere.

For example, if you take a sphere, then the derivative of the polar basis vector with respect to the polar coordinate is in the radial direction. How can something in the radial direction be written as a linear combination of vectors in a plane perpendicular to the radial direction?

[tex]\partial_i e_j=\Gamma^{k}_{ij} e_k [/tex]

Or is this a misunderstanding of the Christoffel symbols? The above equation seems to make sense in spherical coordinates in 3-dimensional space, but not in spherical coordinates on a 2-dimensional sphere.