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fresh_42

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No. There is nothing special about scalar or vector fields. They are just another way to look at functions, either scalar valued functions, or vector valued functions.I thought that there was a method to control each pair (x;y) in the two fields using the same couple in order to demonstrate the AM GM inequality

Now why they are considered at all? This comes from the fact that they occur in physics all the time: measurements at the locations of the domain (phase space), and the change of these measurements if you follow a path (flow) through these fields. The vector fields are typically tangent spaces at a certain location, collected in a set over all possible locations. E.g. the path of an orbiter through a gravitational field is such a flow; same as car driving, you always follow the tangent vectors until you apply a force to change direction. Vector fields are thus a convenient notation to handle all these different tangent spaces (one at each location) at the same time and describe flows through such fields.