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mordechai9

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First we consider arbitrary smooth functions. Smooth functions are continous, with continuous derivatives of all orders. We consider these functions as mapping from some open set U to R, in which case it is the typical scalar function. Otherwise, if the function maps from U to G, where G has dimension equal to U, then we have a vector field.

For the vector field, consider the domain and image, with the standard meaning. Then each level set of the domain of the vector field can be parametrized by a smooth curve. If this curve has a velocity vector everywhere equal to the image of the vector field, then this curve is an integral curve. By the existence and uniqueness theorem we can show that there exists a maximal integral curve for each level set of the vector field.

Next we go back to the consideration of smooth scalar fields. Then it can be shown that gradient of the scalar field at some point p on the curve is orthogonal to all vectors tangent to the integral curve at the point p. Conversely we can show that if a vector is tangent to the integral curve at point p, then it is orthogonal to the gradient of the scalar field at that point. So we can establish a tangent space, meaning the subspace at point p of all vectors tangent to the gradient of the scalar field. Note that this subspace, the tangent space, has dimension value that is one less than the dimension of the domain.

Next we consider surfaces as level sets of smooth scalar fields whose gradient at all points on the surface is nonzero, so that each point on the surface can be associated with a unique tangent space.

This is my brief conceptual summary... next chapters are, "Vector fields on surfaces, orientation", "the gauss map", "geodesics", "parallel transport"...