Tangent Vectors and Vector Fields
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Tangent Space
The set of all tangent vectors that have the same point of application, p, together form a vector space, which is called the tangent space, [Equation goes here - download the original pdf to see it.]. For a Euclidean space, [symbol], the tangent space for each point [Equation goes here - download the original pdf to see it.] is also isomorphic to [symbol]; to make this explicit we may use the symbol, [Equation goes here - download the original pdf to see it.] for this tangent space. (That is, given a fixed point, p, the mapping [Equation goes here - download the original pdf to see it.] is a linear isomorphism - i.e. one-one and surjective.) Each tangent space [Equation goes here - download the original pdf to see it. has dimension three, and a basis of vectors, [Equation goes here - download the original pdf to see it.] , so the tangent vector [Equation goes here - download the original pdf to see it.] can be written in component form [Equation goes here - download the original pdf to see it.] All these notations are equivalent, and, of course, very often the point p is suppressed. Here [Equation goes here - download the original pdf to see it.] act in the same way as [symbol] do for normal vectors; they are unit basis vectors pointing in the same direction as the x-, y- and z- axes respectively Of course, alternatively, we may wish to be explicit about what point p and write [Equation goes here - download the original pdf to see it.] Note, here we have been dealing with tangent spaces of dimension three [Equation goes here - download the original pdf to see it.]. Clearly, we could deal with tangent spaces of dimension two, or, more abstractly generalise to spaces of higher dimension.
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Contents of
Tangent Vectors and Vector Fields
1 Properties of the line integral
2 Vector Field
3 Tangent Vectors
4 Tangent Space
5 Addition and Scalar Multiplication of Tangent Vectors
6 Vector Fields and Tangent Spaces
7 Vector field lines
8 Differentiable Vector Field
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