Math 252- 15.1: Vector Fields

Front 1

Define a Vector Field both over a plane region R and over a solid region Q in space

Back 1

Over a Plane region R:
is a function F that assigns a vector F(x,y) to each point in R.

Over a solid region Q in space:
a function F that assigns a vector F(x,y,z) to each point in Q.

Front 2

The Procedure to Sketch a Vector Field in the Plane:

Back 2

1. Best to plot vectors of equal magnitude
- must find level curves in scalar fields, therefore set the magnitude of the vector field function equal to a constant, and take a look at the resulting equation

2. Make table w/ two columns
column #1:points (x,y)
column#2: the vector result (output) of the vector field you are attempting to sketch.

3. plot the vectors at specified point
- maintain both magnitude ant direction

Front 3

What is a conservative vector field?

Back 3

A vector field F that has an existing differentiable function f such that F= the gradient of f. The function is called the potential function for F.

Front 4

Test for Conservative Vector Field in the Plane:

Back 4

meaning you can recover the function for which this vector field is a gradient (potential function)

Front 5

How to find a Potential function for F(x,y):

Back 5

1. From F(x,y) perform the test for conservative vector field, if it is conservative continue if not then there does not exist a potential function

2. need to anti-differentiate twice/3 times, once w/ respect to x, w/ respect to y and if applicable w/ respect to z.

3.sum up the results, if repeats occur only need to take once.

Front 6

Definition of Curl of a Vector Field:

Back 6

Note: if the curl F=0, then F is said to be irrotational and conservative

Also the curl can only be found in vector fields in space.

Front 7

Test for Conservative Vector Field in Space:

Back 7

Front 8

Finding the Divergence of a Vector Field:

Back 8