11.16 - Section 13.1: Vector fields (concluded) Group Work:...

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Section 13.1: Vector fields (concluded) Group Work: Sketching vector fields, 1 and 2 only (page 7 of Instructor’s Guide) Section 13.2: Line integrals Key ideas? ..?. . It’s a misnomer: the “lines” are actually paths Line integral of a scalar field w.r.t. arc-length o Definition as a limit of Riemann sums: formula 2 o Geometric meaning: Figure 2 o How to compute it: Example 2 Line integral of a scalar field w.r.t. coordinate variables o C f ( x , y ) dx o C f ( x , y ) dy o C P dx + Q dy Line integral of a vector field o Definitions as a limit of Riemann sums o Geometric/physical meaning o How to compute it Vector fields and work: W = C F d r
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f is non-negative, then C f ( x , y ) ds can be interpreted as the area of a curved wall whose base follows the curve C and whose height above the base-point ( x , y ) is f ( x , y ). How can we write the line integral of a scalar field (with respect to arc length) in terms of an ordinary first-year calculus integral? ..?. . Formula (3): C f ( x , y ) ds = a b f ( x ( t ), y ( t )) sqrt(( x ( t )) 2 +( y ( t )) 2 ) dt What are a and b ? ..?. . They arise from a parametrization of the curve C as the set of points ( x ( t ), y ( t )) as t goes from a to b . Isn’t there more than one way to parametrize the curve C in this way? Which parametrization should one use? ..?. . Whichever parametrization is most convenient, because it doesn’t matter which one you use ; formula (3) will give the same answer no matter what parametrization one uses, as long as… ..?. . every point along
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11.16 - Section 13.1: Vector fields (concluded) Group Work:...

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