Consider the vector function F 2e2x sinyi e2x cosyj Show t

Consider the vector function F = 2e2x sin(y)i + e2x cos(y)j. Show that F is a conservative vector field. [4 marks] Find a scalar function f(x, y) so that F = f. [4 marks] Hence (or otherwise) evaluate where C is the semi-circle of radius 1 taken anticlockwise from (1,0) to (-1,0). [4 marks]

Solution

a)d(Work done)=F.dr=2e^2xsinydx+e^2xcosydy=d(e^2xsiny) Hence it is a conservative field as it id dependent on only initial and final positions b)F=deltaf=delta(e^2xsiny) c)W=integration(d(e^2xsiny)) from (1,0) to (-1,0) =0 Ans