Lecture: Section In this section, we will consider more rigorously the question of whether a vector field is path dependent.

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1 Lecture: Section 184 Path Dependent Fields and Green s Theorem In this section, we will consider more rigorously the question of whether a vector field is path dependent Graphical Interpretation We know that a vector field is path independent if, and only if, closed curve C One way to test this condition is graphically Example: Consider the two vector fields shown below 0 for every Y 0 Y X X We could consider a circular path on Figure I and conclude that the circulation is non zero This is conclusive in proving that the field is pathdependent A square path on Figure II is effective for showing that a zero circulation should be expected for this field (however, this is not conclusive) Algebraic Determination of Path Dependence We already presented a theorem in Section 183 to test for path independence in 2 D vector fields We will give a bit more background here and extend the concept to 3 D T King: MA3160 Page 1

2 Lecture 184 Given a vector field, if is conservative, ie, a gradient field that is path independent, then there is a potential function f such that If a function f does not exist, then is path dependent As before, we observe: Now, assuming f has continuous partial derivatives, by the equality of mixed derivatives We can also write the equation above in the following from when is a gradient field Note that the quantity on the left hand side is called the scalar curl of the 2D vector field We will find out in Chapter 20, that the curl is related to the speed of rotation (ie, angular velocity) of the field We can then infer that rotational vector fields are, by definition, path dependent 0 Scalar Curl of It is very important, at this point, to mention that the curl is a vector property In the equation above, what is given is the magnitude of the curl (ie, scalar curl) We will find out later that the curl vector is perpendicular to the vector field which, in this case would be in the z direction Further, the equation is written as shown above in order to satisfy convention, that is, a positive curl implies counterclockwise rotation when the right hand rule is used The Curl Test for Vector Fields in 3 D Space There is an analogous test for vectors in 3 D space, although we don t yet have all the tools to justify it (again in Chapter 20) So we will state the test as a theorem without proof T King: MA3160 Page 2

3 Lecture 184 The curl in 3 D is given as: If curl 0, then the is path independent We also note that the result of this computation will be a vector Example: 18Review Exercise 23 Check if the following vector field is path independent is path independent Physical Interpretation of the Curl So what does the curl represent? 0 We will drop back to 2 D for a minute to try to get our arms around the concept Think of dropping a small paddlewheel into the fluid vector field An excellent applet showing this is available at the link from Cornell on my webpage The curl is a vector for which the magnitude is related to the speed of rotation of the wheel and the direction is along the axis of rotation ( direction for rotation in the xyplane use the right hand rule) Consider the vector field 11 2 T King: MA3160 Page 3

4 Lecture 184 Now consider 21 Note that for the first field, the curl is constant over the field, but for the second field, the curl depends upon the position in the field Green s Theorem We now are ready to discuss Green s Theorem This theorem will give us another way to determine the circulation of a vector field (ie, instead of evaluating the line integral) A conceptual way of thinking about this would be to sum the circulations within the region, across the area of the region Now consider this situation with the region split into grids each with its own microcirculation This is depicted in the figure below We see that the circulations cancel on all the inner boundaries, and the circulation is determined by the components on the surface Hence integrate the microcirculation over the area (a double integral) this is be equivalent to summing them along the T King: MA3160 Page 4

5 Lecture 184 boundary of the surface (a line integral over path C) Mathematically, this can be written as: "" But we know from out recent discussion that the curl gives us the microcirculation so: Note that this will be a double integral and that there are a couple of restrictions: 1 c is a piecewise smooth simple closed curve that is the boundary of an open region 2 We move around the path (counterclockwise) so that the region is on our left 3 The region is a plane (2 D) Example: 18Review30 Calculate the line integral of along the perimeter of a triangle of area = 7, moving counterclockwise 1 Determine if the field is conservative Use Green s Theorem to calculate the circulation of F Clearly, this is easier than using the line integral! Final Thought So curl is a measure of circulation per unit area T King: MA3160 Page 5

MATH 280 Multivariate Calculus Spring Derivatives of vector fields: divergence and curl

MATH 280 Multivariate Calculus Spring 2011 Vector fields in the plane Derivatives of vector fields: divergence and curl Given a planar vector field F P x, y î + Qx, y ĵ, we can consider the partial derivatives.