Question

Consider the following region R and the vector field F a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in the circulation form of Green's Theorem and check for consistency. c. State whether the vector field is conservative. F-3y,3x); R is the triangle with vertices (0, 0), (1, 0), and (0, 1) a. The two-dimensional curl is D (Type an exact answer, using π as needed.) b. Set up the integral over the region R. dy dx Set up the line integral for the line segment between (0, 0) and (1, 0) 0 Write the line integral for the line segment between (1, 0) and (0, 1 0 Write the line integral for the line segment between (0, 1) and (0, 0) Io Evaluate these integrals and check for consistency O A. The integrals are consistent because they both evaluate to OB. The integrals are not consistent because the integral over R evaluates to while the line integral evaluates to c. Is the vector field conservative? OA. Yes, since the line integral of the vector field is around the given region, the line integral of the vector field is around any region. OB. No, while the line integral of the vector field is around the given region, the line integral of the vector field will be different around a different region. O C. Yes, since the curl of the vector field is the line integral of the vector field over every simple closed piecewise-smooth oriented is OD. No, since the curl of the vector field is the line integral of the vector field over every simple closed piecewise-smooth oriented is not

Answer 1

a. The two-dimensional curl is (Type an exact answer, using r as needed.) b. Set up the integral over the region FR Set up the line integral for the line segment between (0, 0) and (1, 0). 0 Write the line integral for the line segment between (1, 0) and (0, 1) 3dt 0 Write the line integral for the line segment between (0, 1) and (0, 0). dt

Evaluate these integrals and check for consistency A. The integrals are consistent because they both evaluate to 3 OB. The integrals are not consistent because the integral over R evaluates towhile the line integral evaluates to c. Is the vector field conservative? OA. Yes, since the line integral of the vector field is around the given region, the line integral of the vector field is around any region. OB. No, while the line integral of the vector field is around the given region, the line integral of the vector field will be different around a different region OG. Yes, since the curl of the vector field is the line integral of the vector field over every simple closed piecewise-smooth oriented is OD. No, since the curl of the vector field is6 the line integral of the vector field over every simple closed piecewise-smooth oriented is not

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