Determine wether the given vector field is conservative If i

Solution

Suppose F = F(x,y, z) is a gradient field with F = Ñf , S is a level surface of f, and C is a curve on S. What is the value of the line integral_C F·dr? Zero. If you follow a level surface, the gradient and hence F is everywhere perpendicular to the curve F*dr = 0 so the integral must be zero 2- Consider the vector field F(x,y) = (2x2 +y2,2xy). Compute the line integrals_C1 F· dr and Integral_C2 F· dr, where C1 is the curve r1(t)=(t, t2) for 0t 1 andC2 is the curve r2(t)=(t, t), also forR0 <= t <= 1. Integral_C1 F·dr = . Int[(i(2x2+y2)+j(2xy))*(idx + jdy)] = Int[(2x2+y2)dx +(2xy)dy)] Here x=t, y=t2 dx = dt dy = 2tdt Integral_C1 F·dr = Int(t=0 to 1)[(2t2+(t 2)2)dt +(2tt2)2tdt)] = Int(t=0 to 1)[(2t2+t 4 + 4t4)dt] = Int(t=0 to 1)[(2t2+5t 4dt] = ((2/3)t3 + t5)(t=1 - t=0) = 5/3 Integral_C2 F·dr = Int[(i(2x2+y2)+j(2xy))*(idx + jdy)] = Int[(2x2+y2)dx +(2xy)dy)] Here x=t, y=t dx = dt dy = dt Integral_C1 F·dr = Int(t=0 to 1)[(2t2+ t2)dt +(2tt)dt)] = Int(t=0 to 1)[(5t2dt] = ((5/3)t3)(t=1 - t=0) = 5/3 Same value