Homework Soursea Verify that the force eld Fx y cos xcos y i x siny j is
(a) Verify that the force eld F(x, y) = (cos x-cos y) i + x siny j is conservative.
(b) Find a potential function phi(x, y) for F(x, y).
(b) Find the work done by the force field F(x, y) along the curve x = t/2,
y = (pi-t)/2, 0 ? t? pi
(b) Find a potential function phi(x, y) for F(x, y).
(b) Find the work done by the force field F(x, y) along the curve x = t/2,
y = (pi-t)/2, 0 ? t? pi
Solution
(a)curl F=0
i j k
d/dx d/dy d.dz
cosx-cosy xsiny 0
=i(0(+j(0)+k(siny-siny)
=0
so field is conservative
(b)dV/dx=Fx=cosx-cosy
Vx=sinx-xcosy
dV/dy=Fy=xiny
Vy=-xcosy
V=(sinx-xcosy)i-xcosyj
(b)dW=F.dr
dx=1/2
dy=-1/2
F.dr=1/2(cos(t/2)-sin(t/2))+t/4cos(t/2) dt
now integrating this we get
W=0.571
