1. 16.3 Vector Fields Understand the concept of a vector field
Determine whether a vector field is conservative
Find the curl of a vector field
Find the divergence of a vector field

4. Note: Although a vector field consists of infinitely many vectors, you can get a good idea of what the vector field looks like by sketching several representative vectors F(x, y) whose initial points are (x, y).

6. Solution:

9. Vector Fields in 3D

11. Solution:

12. Testing for conservative vector fields in the plane

14. Solution:

15. Your turn…

17. Solution:

18. Since this is a dot product, you will end up with a SCALAR field.
If the div F=0, then F is said to be divergence free.
Note: Divergence measures the rate of particle flow per unit volume at a point.

20. Since this is a cross product, you should end up with a VECTOR field.
*We did the dot product to find the curl in Section This method uses determinants
Since this is a cross product, you should end up with a VECTOR field.
Moreover, if curl F=0, we say that F is irrotational

22. Test for Conservative Vector Field in Space
A vector field is conservative iff curl F(x, y, z)=0 Once a vector field has passed the test for being conservative, you can find the potential function by taking 3 separate integrals

23. Putting it altogether…

25. Applications
Velocity Fields (ex. Wheel rotating on an axle—the farther a point is from the axle, the greater the velocity
Gravitational fields=force of attraction exerted on a particle of mass aka. Newton’s law of gravitation (ex. Central force field)
Electric force fields aka. Couloumb’s Law=force exerted on a particle with electric charge

26. Contour maps: gradient points in the direction that is the steepest uphill on the surface (the gradient field and the contour map should be orthogonal to each other)

27. Conservative Vector Fields and Independence of Path
Understand & use Fundamental theorem of line integrals
Understand the concept of independence of path
Understand the concept of conservation of energy

28. Goal…

32. Fundamental theorem of Calculus for line integrals

34. Solution: