Vector fields, integrals, and Green's Theorem

Need full steps and solutions shown for questions 21 and 24 only.

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• A curve is called closed if its terminal point coincides with its initial point: r(b) = r(a). Open means that for function D, for every point, P in D, there is a disk with center P that lies entirely in D (so D doesn't contain any of it's boundary points). Connected means that any two points in D can be joined by a path that lies in D. A simply connected region in the plane is a connected region D, such that every simple closed curve in D encloses only points that are in D.

• Intuitively speaking, a simply connected region contains no hole and cant consist of two separate pieces. If you can sketch the region, I can determine the answer based on examples given for the concepts for question 9.

• For 17 part b, grid curves obtained by letting u be constant are horizontal lines, whereas grid curves with v constant are circles. If v is constant, the sin v and cos v are constant, so the parametric equations resemble a helix. This, grid curves with v constant are spiral curves. Grid curves with u constant are curves that look like circles.

• Also, three out of the first five questions came up as incorrect, so can you please double check all the answers.

• I will recorrect everything, let me know the exact question numbers which are wrong!

• If you have the second try to submit the answers for wrong entries ?

• Sketch explanation added in the solution for #9.

• Questions 1, 3, and 5 were wrong. If you could correct those and double check the rest of the questions to ensure they are correct, it would be appreciated.

• Corrections applied

• Thank you. Everything looks good, I'm just not sure about 17 b and c. I tried guessing (based on my notes) the 2nd option for u constant and the 3rd option for v constant, but both were wrong. I'm not sure if you'd have any guesses based on what I shared.

• If you have any question which is similar to 17 then attach it here by clicking add a file! I may easily do after that!

• It doesn't let me attach more files, but no worries.

• Thank you,