True or False? Justify your answer with a proof or a counterexample.
Vector field is conservative.
For vector field if in open region then
The divergence of a vector field is a vector field.
If then is a conservative vector field.
Draw the following vector fields.
Are the following the vector fields conservative? If so, find the potential function such that
Evaluate the following integrals.
along from (0, 0) to (4, 2)
where S is surface
Find the divergence and curl for the following vector fields.
Use Green’s theorem to evaluate the following integrals.
where C is a square with vertices (0, 0), (0, 2), (2, 2) and (2, 0) oriented counterclockwise.
where C is a circle centered at the origin with radius 3
Use Stokes’ theorem to evaluate
where is the upper half of the unit sphere
where is the upward-facing paraboloid lying in cylinder
Use the divergence theorem to evaluate
over cube defined by
where is bounded by paraboloid and plane
Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb.
Find the total mass of a thin wire in the shape of an upper semicircle with radius and a density function of
Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for with a density function
Use the divergence theorem to compute the value of the flux integral over the unit sphere with