Where the curve is described by r t x t y t Arc length Lecture 19 Integration on a curve Example f x y Vector eld in R V M Nj M N where M and N are functions of x and y Ways a vector eld arises Gradient Velocity eld wind or uid current Force eld gravity or electrostatic Examples an eastward breeze V 5 big bang V x yi rotating V y xj Force eld Work If an object is moved through a force then work is done W F r where r is displacement and where F is constant Note if movement is perpendicular to the force no work is done What if F changes W F r r W F r r F dr Example F y x C r t t 0 t 1 We want to nd W F dr Example F M N r x y Note We could have started with f x or f y which are different from one another How to compute these r t t 0 t 1 what is y x Another way to write it r t r x y T ds x y where T is the unit tangent and s is the arc length measure Example f x y c r 4t 1 3t 1 1 t 1 We want to nd s x y
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