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 Emeritus Sci Advisor PF Gold P: 9,010 The tangent space $$T_pM$$ of the manifold M at point p can be defined as the vector space spanned by the basis vectors $$\frac{\partial}{\partial x^\mu}\bigg\lvert_p$$ where x is a coordinate system (a chart). (There is also a coordinate independent definition, but it will result in the same quantities being called tangent vectors, so it's equvalent to this one). Let F denote the set of smooth ($$C^\infty$$) functions from M into R (the set of real numbers). The vectors in the tangent spaces are linear functions from F into R In the physics literature, a vector field is sometimes defined as a function that takes each point p in a subset U of M to a vector in the tangent space at p. In the mathematics literature the definition is a little more complicated, but it's still pretty close to the sloppy physicist's definition. I'm not 100% sure that I remember the mathematical definition 100% correctly, but I think this is at least very close to it: The tangent bundle TM of the manifold M is defined by $$TM=\big\{(p,v)|p\in M, v\in \bigcup_{q\in M}T_qM\big\}$$ The function $$\pi:TM\rightarrow M$$ defined by $$\pi(p,v)=p$$ is called the projection. A vector field is a section of the tangent bundle. A section is a function $$X:M\rightarrow TM$$, such that $$\pi(X(p))=p$$