Tangent Vectors and Vector Fields
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Vector Field
In general a vector field, V, is a mapping (function) from one vector space to another. For example, suppose we represent an ocean as a two-dimensional disk and the direction and magnitude of the current on the surface of the ocean by a two dimensional vector, then the function that assigns to each point on the surface of the ocean its current vector is a vector field. [Diagram goes here - download the original pdf to see it.] This is the vector field [Equation goes here - download the original pdf to see it.] This is an assignment from the point [Equation goes here - download the original pdf to see it.] to the vector with direction [Equation goes here - download the original pdf to see it.] at the point p. This makes it clear that a vector field is composed of vectors that have two parts (i)A point of application, p (ii)A vector v A regular vector has only the vector part, and to distinguish the new kind of vector from the "regular" one, it is called a tangent vector and is designated [symbol]
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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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