Tangent Vectors and Vector Fields
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Vector Fields and Tangent Spaces
A formal definition of a vector field, [Equation goes here - download the original pdf to see it.], is that it is an assignment of tangent vectors to points of Euclidean space. Equation goes here - download the original pdf to see it. Explicitly, [Equation goes here - download the original pdf to see it.] We have already indicated, above, that the functions, [Equation goes here - download the original pdf to see it.], are themselves (here, three-dimensional) scalar fields. A vector field may be visualised as a collection of arrows, one for each point of Euclidean space. At each point each arrow is just one of the tangent vectors belonging to the tangent space for the given point. We can add two vector fields, V and W, by adding their values at every point. [Equation goes here - download the original pdf to see it.] This principle is called pointwise addition and may be extended to cover the notion of the scalar multiplication of vector fields thus [Equation goes here - download the original pdf to see it.] We have already indicated that the tangent space to a point, p, has a basis [Equation goes here - download the original pdf to see it.] Since for every point, p, these basis vectors are identical, then p is redundant [Equation goes here - download the original pdf to see it.] This is why the vector field that assigns to each distinct point a unique vector may be broken down into components [Equation goes here - download the original pdf to see it.] The [Equation goes here - download the original pdf to see it.] notation is often useful for summarising results. That is, for example, the law of addition of vector fields and law of scalar multiplication may be expressed. [Equation goes here - download the original pdf to see it.] Sometimes, also, for convenience, we use the symbols [symbol] instead of [Equation goes here - download the original pdf to see it.] for the basis vectors to the tangent space at p. Perhaps [symbol] should be reserved for the basis of the Euclidean space in which p is found and [Equation goes here - download the original pdf to see it.] for the tangent space in which [Equation goes here - download the original pdf to see it.] is found . However, as the two spaces are in fact isomorphic (that is, "identical"), the interchange of notations is often convenient, as in the following example. Example Newton's law of gravitation gives rise to a vector field describing the gravitational force acting on a particle within a gravitational field. Suppose a particle of mass M is fixed at a point P0 and that another smaller particle of mass m is situated at a point P in space. [Example goes here - download the original pdf to see it.]
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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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