Tangent Vectors and Vector Fields
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Tangent Vectors
As already indicated a normal vector is an orientated line segment in space given by its size (magnitude) and direction. It is this definition that has so many applications in mechanics because it enables us to regard any two vectors with the same magnitude and direction as the same regardless of their point of application. Forces are vectors, and it is this definition that enables us to add forces to form a resultant force. Actually, it is assumed that the point of application of the force does not make a difference to the resultant, but when forces are applied to a rotating object this is not the case; different forces at different points of a rotating dumbbell, for instance, produce different rotational effects (torque), so this already shows that the point of application cannot always be ignored. So when it comes to the effect of forces, over, for example, surfaces, then the definition of a vector must be supplemented to include both the vector part and the point of application. This is the tangent vector, [Equation goes here - download the original pdf to see it.] [Diagram goes here - download the original pdf to see it.] As the diagram indicates, we picture the tangent vector [Equation goes here - download the original pdf to see it.] as an arrow from the point P, with vector p to the point Q, with vector p + v. It follows from the definition of a tangent vector that two tangent vectors are equal if, and only if, they have the same point of application and the same vector part; that is [Equation goes here - download the original pdf to see it.] Two distinct tangent vectors are said to be parallel if they have the same vector part but different points of application.
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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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