Green's Theorem

Green's theorem relates a line integral around a simple closed curve C to a:

• triple integral over the region enclosed by C

• double integral over the region enclosed by C

• surface integral over the region enclosed by C

• line integral over the region enclosed by C

Which of the following is NOT a condition for applying Green's theorem?

• C is a positively oriented, piecewise smooth, simple closed curve in the plane.

• R is the region bounded by C.

• Functions f and g have continuous partial derivatives on an open region that contains region, R.

• The vector field F = <f g> is conservative.

The circulation of a vector field F around a simple closed curve C is calculated by:

• the line integral of F over C

• the double integral of the curl of F over the region enclosed by C

• the surface integral of F over the region enclosed by C

• the line integral of the divergence of F over C

The curl of a vector field F = <P, Q> in the plane is given by:

• ∂P/∂x + ∂Q/∂y

• ∂Q/∂x - ∂P/∂y

• ∂P/∂y - ∂Q/∂x

• ∇ • F

If the curl of a vector field F is zero in a simply connected region, then:

• F is a scalar field

• F is not conservative

• Green's theorem cannot be applied to F

• The circulation of F around any closed curve in the region is zero