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

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).

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Vector Fields in 3D

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Testing for conservative vector fields in the plane

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Your turn…

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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.

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 16.2. 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

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

Putting it altogether…

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

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)

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

Goal…

Fundamental theorem of Calculus for line integrals

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