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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 33 Line Integrals, conservative vector fields, Curl and potential functions in 3D April 30 Reading Material: From Simmons : 21.5. From Lecture Notes : V8, V11, V12. Last time: Divergence Theorem Today: Line Integrals, conservative vector fields, Curl and potential functions in 3D. 2 Line Integrals in 3D In this section we recall how we write line integrals for a 3D vector fields, the theorem about conservative vector fields and the fundamental theorem of line integrals. All three of these facts are just a generalization of what we know already in 2D. In this lecture we assume that all vector fields are continuously differentiable. Line Integrals in 3D : Assume ~ F = M i + N j + P k and that a curve C is parametrized by the vector position ~ r ( t ) = x ( t ) i + y ( t ) j + z ( t ) k for a t b then Z C F d~ r = Z C M dx + N dy + P dz = b Z a M ( x ( t ) , y ( t ) , z ( t )) x ( t ) dt + N ( x ( t ) , y ( t ) , z ( t )) y ( t ) dt + P ( x ( t ) , y ( t ) , z ( t )) z ( t ) dt. Exercise 1. Consider the vector field ~ F = y i- x j + z k and the curve C given by ~ r ( t ) = cos t i + sin t j + t k for t 4 . Compute the work of ~ F along C ....
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