$Vector fields, integrals, and Green's Theorem$

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$Vector fields, integrals, and Green's Theorem$

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

Multivariable Calculus Vector Fields

Bunbun

229

Matchmaticians Vector fields, integrals, and Green's Theorem File #1

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Please find the solution attached. Need more source or lecture notes regarding question 9 and 17 part b.
Thank you.

Sketch for question 9:
The region will be whole $R^2$ region except all the points on the vertical line at $x=1$. So you can assume the graph as divided into two regions one is left to x=1 line and one to the right. Exclude line points.

$\color{red}{\text{Corrections:}}$
For Q3, I just realised the graphs b and d are almost similar except the directions near x-axis. So, I realised my mistake and correct answer should be d.

For Q1, I am not sure about this one beucase I plotted the same using 3D vector plotter and gave me the result I chose, But if it is not option a, then it must be c. Because the directions can't be in z direction as in parts b and d.

For Q5, I see the curve is not closed I thought it is closed semi circle which is not hence Greens theorem is not applicable, the correct answer has been modified in the attachment which should be $\sin 3-\cos 3+\cos 2$

3 Attachments

Aman R

643

Bunbun

0

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.
Bunbun

0

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.
Bunbun

0

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.
Bunbun

0

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

0

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

0

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

0

Sketch explanation added in the solution for #9.
Bunbun

0

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.
Aman R

0

Corrections applied
Bunbun

0

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.
Aman R

0

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!
Bunbun

0

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

0

Thank you,

$Vector fields, integrals, and Green's Theorem$

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