(1) Suppose vec(\omega ) is the axis of rotation (using the right-hand rule) of a rotating rigid body and that |vec(\omega )| is equal to the angular velocity of the rotation. Then it follows that the velocity at a position vec(r)=(:x,y,z:) in the rigid body is vec(v)=vec(\omega )\times vec(r). Which is the curl of vec(v) ?
(a) vec(\omega )
(c) -vec(\omega ) (e) vec(0)
(b) 2vec(\omega )
(d) -2vec(w)
(f) None of these
(2) Suppose F=(:f,g:) where f,g have continuous first order partial derivatives throughout the entire xy-plane and grad*F has an average value of 2 on the unit disk D={(x,y)inR^(2)|x^(2) y^(2)<=1}. Determine the outward flux of F across the boundary of D. Round to the nearest hundredth.
(3) Which of the following iterated double integrals is equal to the outward flux of the vector field F=(2xy,x y^(2)) on the boundary of the region in the first quadrant of the xy-plane bounded by x^((1)/(3)) y^((1)/(3))=1 ?
(a) \int_0^1 \int_0^(1-x^((1)/(3))) 4ydydx (d) \int_0^1 \int_0^(\root(3)(1-x^(3))) 1-2xdydx
(b) \int_0^1 \int_0^((1-x^((1)/(3)))^(3)) 1-2xdydx (c) \int_0^1 \int_0^(1-x^((1)/(3))) 1-2xdydx
(c) \int_0^1 \int_0^((1-x^((1)/(3)))^(3)) 4ydydx (f) \int_0^1 \int_0^(1-x) 4ydydx
(4) A point charge in a plane generates the radial electric field in that plane given by E=(50)/(r^(3))r in SI units (SI unit volts per meter - kilogram meter per second ^(3) per ampere) where r=(:x,y:) is the position relative to the point charge and r=|r|. Using o\int_(delD) E*Nds=∬_(D)vec(grad)*EdA, calculate the electric flux in SI units (volt-meters) out of the boundaries delD of the annulus D with inner radius 1 and outer radius 2 (centered at the point charge). Round to the nearest tenth.
(5) Suppose we have the time-dependent magnetic field vec(B)(t)=(e^(-(t)/(10)))/(x^(2) y^(2))(:-y,x,0:) (SI unit tesla - kilogram per second squared per ampere) and a time-dependent electric field vec(E)(t)=E(x,y,t)hat(z) (SI unit volts per meter-kilogram meter per second ^(3) per ampere) in the z-direction that is independent of z at time t>=0 seconds. One of Maxwell's equations states that vec(grad)\times vec(E)=-(del(vec(B)))/(delt). Determine the z-component of the electric field at ( 1,1,1 ) at time t=10 given that the electric field vanishes at ( 1,0,0 ). Round to the nearest thousandth.