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Understanding Vector Fields: Definitions, Properties, and Applications

A vector field is a function that assigns a two-dimensional vector to every point in the xy-plane, expressed as F(x,y) = P(x,y)i + Q(x,y)j. It can also be extended to three dimensions. Vector fields are pivotal in physics, representing quantities like fluid velocity and gravitational forces. This guide explores concepts such as conservative fields, potential functions, curl, and divergence, providing examples and theorems to illustrate these properties. Learn how to determine whether a field is conservative and investigate its characteristics further.

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Understanding Vector Fields: Definitions, Properties, and Applications

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  1. Vector Fields Def. A vector field is a function F that assigns to every point in the xy-plane a two-dimensional vector F(x,y). F(x,y) = P(x,y)i + Q(x,y)j These are the same as vector-valued functions, but now they are multi-variable. We can also have vector fields with three components, graphed in 3-space.

  2. Ex. Sketch the vector field F(x,y) = -yi + xj.

  3. Ex. Sketch the vector field F(x,y,z) = zk.

  4. Vector fields can be used in some areas of physics: • As fluid flows through a pipe, the velocity vector V can be found at any location within the pipe. • The gravity created by a mass creates a vector field where every vector points at the mass, and the magnitudes get smaller as points are chosen further from the mass.

  5. We’ve already worked with an example of a vector field… GRADIENT Ex. Let f (x,y) = 16x2y3, find f.

  6. Def. A vector field F is called conservative if there is some function f such that F = f. The function f is called the potential function of F.

  7. Ex. Show that F(x,y) = 2xi + yj is conservative.

  8. Thm. The vector field F(x,y) = P(x,y)i + Q(x,y)j is conservative if and only if

  9. Ex. Determine if the vector field is conservative. • F = x2yi + xyj • F = 2xi + yj

  10. Ex. Show that F(x,y) = 2xy3i + (3x2y2 + 2y3)j is conservative, then find its potential function.

  11. Curl and Divergence Def. The curl of F(x,y,z) = Pi + Qj + Rk is Ex. If F = xzi + xyzj – y2k, find curl F.

  12. Thm. A vector field F is conservative if and only if curl F = 0. So the last vector field was not conservative. If curl F = 0 at a point, then we say that the vector field is irrotational at the point.

  13. Ex. Show that is conservative, and find its potential function.

  14. Def. The divergence of F(x,y,z) = Pi + Qj + Rk is Ex. If F = xzi + xyzj – y2k, find div F.

  15. Note: curl F is a vector div F is a scalar Thm. div (curl F) = 0

  16. Ex. Show that F = xzi + xyzj – y2k can’t be written as the curl of another vector field, that is F≠ curl G.

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