Section 18.4Path-Dependent Vector Fields and

Green’s Theorem

How can we tell if a vector field is path-dependent?

• Suppose C is a simple closed curve (i.e. does not intersect itself)

• Let P and Q be two points on the curve• Let C1 be a path from P to Q in one direction and

C2 be a path from P to Q in the other direction

• Then

21 CC

rdFrdF

02121

CCCCCrdFrdFrdFrdFrdF

• Thus a vector field is path-independent if and only if for every closed curve C

• So to see if a vector field is path-dependent we look for a closed path with a nonzero integral

• This can also be done algebraically• Let’s see if we can find a potential function for

C

rdF 0

xyxyF ,2

We can check by doing the following• Let be a vector field• Then it is a gradient field if there is a potential

function f such that

so

then by the equality of mixed partial derivatives we have

21, FFF

dy

df

dx

dfFFF ,, 21

dy

dfF

dx

dfF 21 and

dx

dF

dydx

fd

dxdy

fd

dy

dF 222

1

This gives us the following• If is a gradient field with

continuous partial derivatives, then

• is called the 2-dimensional or scalar curl for the vector field

• Let’s show that is not a gradient field

21, FFF

012 dy

dF

dx

dF

dy

dF

dx

dF 12

xyxyF ,2

Green’s Theorem• Suppose C is a piecewise smooth closed curve that

is the boundary of an open region R in the plane and oriented so that the region is on the left as we move around the curve. Let be a smooth vector field on an open region containing R and C. Then

• Let’s take a look at where this comes from• What is the line integral of on the

triangle formed by the points (0,0), (1,0), (0,1), (0,0)

21, FFF

RCdxdy

dy

dF

dx

dFrdF 12

xyxyF ,2

• Recall this example from last class

and C is the triangle joining (1,0), (0,1) and (-1,0)– Since our path is closed and the region enclosed

always lies on the left as we traverse the path, Green’s theorem applies

– Let’s set up a double integral to evaluate this problem

jyxixyF

)(

Curl Test for Vector Fields in the Plane• We know that if is a gradient field then

• Now if we assume

then by Green’s Theorem if C is any oriented curve in the domain of and R is inside C then

• This gives us the following result

012 dy

dF

dx

dF21, FFF

012 dy

dF

dx

dF

012

dxdy

dy

dF

dx

dFrdF

C R

F

Curl Test for Vector Fields in the Plane• Suppose is a vector field with

continuous partial derivatives such that– The domain of has the property that every closed

curve in it encircles a region that lies entirely within the domain, it has no holes

–

• Then is path-independent, is a gradient field and has a potential function

21, FFF

F

012 dy

dF

dx

dF

F

• Let

• Calculate the partials for this function– Do they tell us anything?

• Calculate where C is the unit circle centered at the origin and oriented counter clockwise– Does this tell us anything?

22),(

yx

jxiyyxF

C

rdF

Curl Test for Vector Fields in 3 Space• Suppose is a vector field with

continuous partial derivatives such that– The domain of has the property that every closed

curve lies entirely within the domain–

• This is the curl in 3 space

• Then is path-independent, is a gradient field and has a potential function

321 ,, FFFF

F

F

0,, 123123

dy

dF

dx

dF

dx

dF

dz

dF

dz

dF

dy

dF

Examples• Are the following vector fields path

independent (i.e. are they gradient fields)– If they are find the potential function, f

3 2 2

,

2 ,3

1 1,

1 1 1, ,

v y y

u xy y x y x

Fx y

qx y xy