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

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

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

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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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)

  7. 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

  8. 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

  9. 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

  10. 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?

  11. 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

  12. 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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