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Is a smooth manifold the same as a differentiable manifold ? Does it have to be infinitely differentiable ? Is 3-D Euclidean space infinitely differentiable ?

Is a vector field in 3-D just the gradient operator ?

I have some notes that say a vector field maps a manifold to itself by differentiating a 1-parameter family of maps. I thought a vector field was basically a tangent vector at every point and so was a different manifold ?