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Section 18.4 Path-Dependent Vector Fields and Green’s Theorem

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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 C 1 be a path from P to Q in one direction and C 2 be a path from P to Q in the other direction Then

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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

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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

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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

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

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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

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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

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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

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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?

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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

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Examples Are the following vector fields path independent (i.e. are they gradient fields) –If they are find the potential function, f

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5.1 Real Vector Spaces.

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