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Section 18.4 Path-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 - PowerPoint PPT Presentation
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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
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