Section 17.4The Flow of a Vector Field

• Often we are concerned with the path an object will travel

• We can use the information from the velocity vector field in order to determine a path

• This will depend on the direction and magnitude of the vector as well as the starting point

• Let’s revisit our flow of the gulf stream example beginning with the vector field and then on to the flow lines

• Does this sound familiar?

• Suppose we have a velocity vector field– We will let t be in seconds and use a 2D field

• Then we have • If

• Then

• Thus the flow of the field is the general solution to the above system of differential equations

)(rFv

))(()( trFtr

),(),,(),()(

and)(),()(

yxgyxfyxFrF

tytxtr

)('),())(),(()),(),(())((

tytxtytxgtytxftrF

• Given the system of differential equations

and some initial condition

we can find the particular flow line through the given field

))(),(()())(),(()(tytxgtytytxftx

0

000 )0(

)0(,)0(

yyxx

yxr

Example• Pg. 857, #4

Sketch the vector field and the flow. Find the system of differential equations associated with the vector field and check that the flow satisfies the system.

• Let’s take a look with Maple

tt betyaetxjyixv )(,)(;