Hi. I have a physics background but I am trying to get to grips with differential geometry and struggling with the abstract nature of it. I have a few questions if anyone can help ? 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 ?