Vector Calculus - Whitman College

16 Vector torFieldsThis chapter is concerned with applying Calculus in the context ofvector fields. Atwo-dimensional Vector field is a functionfthat maps each point (x, y) inR2to a two-dimensional vectorhu, vi, and similarly a three-dimensional Vector field maps (x, y, z) tohu, v, wi. Since a Vector has no position, we typically indicate a Vector field in graphicalform by placing the vectorf(x, y) with its tail at (x, y). Figure shows a represen-tation of the Vector fieldf(x, y) =h x/ x2+y2+ 4, y/ x2+y2+ 4i. For such a graphto be readable, the vectors must be fairly short, which is accomplished by using a differentscale for the vectors than for the axes. Such graphs are thus useful for understanding thesizes of the vectors relative to each other but not their absolute fields have many important applications, as they can be used to represent manyphysical quantities: the Vector at a point may represent thestrength of some force (gravity,electricity, magnetism) or a velocity (wind speed or the velocity of some other fluid).

Thus, work in the vector setting is still “force times distance”, except that “times” means “dot product”. If the force varies from point to point, it is represented by a vector field F; the dis-placement vector v may also change, as an object may follow a …

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  Distance, Vector

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