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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 22 Conservative fields and path independence. Gradient fields e potential functions April 2 Reading Material: From Simmons : 21.2. Last time: Vector fields and line Integrals on the plane. Today: Conservative fields and path independence. Gradient fields e potential functions. 2 Work of a vector field along a curve Given a curve C and a vector filed ~ F ( x,y ) = M ( x,y ) i + N ( x,y ) j , the work of ~ F along C is defined as the line integral W = Z C ~ F d~ r = Z C M dx + N dy. and we call this integral the line (or path) integral of ~ F along C . We also recall how line integrals are converted into ordinary 1D integrals by following these steps: 1. Parameterize C as ~ r ( t ) = ( x ( t ) ,y ( t )) = x ( t ) i + y ( t ) j t t t 1 . 2. Then Z C ~ F d~ r = Z t 1 t M ( x ( t ) ,y ( t )) dx dt + N ( x ( t ) ,y ( t )) dy dt dt. (2.1) 3 Two old examples from last lecture Example 1 : Consider the vector field ~ F =- y i + x j and let C 1 be the quarter of the circle centered at the origin of radius 1 in the first quadrant, in the counterclockwise direction. We also introduce the curve C 2 as 1 Last time we computed Z C 1 ~ F d~ r = 2 and Z C 2 ~ F d~ r = 1 + 1 = 2 Conclusion: In general line integrals depend on the shape of the curves connecting two points in space. Example 2 : Consider the vector field ~ G = y i + x j . We proved that the work of ~ G along C 1 is equal the work of ~ G along C 2 . In fact Claim: All path C from (1 , 0) to (0 , 1) will give R C ~ G d~ r = 0. The vector filed ~ G is in fact conservative according to the following definition:...
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