Final round of questions about Green's Thm

[Calpalned]

Homework Statement

*Questions 3, 5 and 6 have already been answered.
1) On page 1091, am I right to assume that example 4 could be solved using Green's Theorem?
2) How about for example 6, on page 1093, or would we need a higher dimensional analog?
3) What is the meaning of ""ds" in example 1 of page 1064? How is ds related to dx and dy? Intuitively, what is the difference between example 1 and example 4?
4) Does the Fundamental Theorem for Line Integrals only work for vector fields? If so, should I automatically use the Fund Thm of Line Integrals for vector fields?
5) The only difference between a line integral and a vector field is the use of $\vec i$ and $\vec j$ in place of dx and dy. Is $\int_C F \cdot dr$ a line integral, or is it a vector? If it's a vector, is it a field or a function?
6) When solving a line integral, how do I know when to use Green's Thm, the Fund Thm of Lne int, or the classic method of using parameters? Would I be told what to do on the exam?
7) On page 1082, question 15 is conservative. Solving using the Fund Thm of Line Int gives 77. Using Green's theorem, the answer is zero. Why is there a difference?
8) Why is example 4 of page 1112 "not simple" while exercise 10 of page $1090$ is?
9) What is the difference between "closed" "simple" and "conservative"?

2. Pictures of my textbook
Attached

Click to enlarge the above: 1091, 1093, 1112, 1114

The Attempt at a Solution

Thank you.

 

[STEMucator]

1) As long as you follow the criterion of Green's theorem, it can be used. Sometimes it's just easier to evaluate the line integral directly though.

2) I'm almost certain I mentioned to you in a prior thread you can only apply Green's theorem in two-dimensions.

4) The fundamental theorem for line integrals is only applicable for a conservative vector field $\vec F = \vec{\nabla f}$.

5) $\oint_C \vec F \cdot d \vec r$ is the line integral of a vector field. With $\vec r(t) = x(t) \hat i + y(t) \hat j + z(t) \hat k$ parametrizing the curve, consider:

$$\oint_C \vec F \cdot d \vec r = \oint_C \left[P \hat i + Q \hat j + R \hat k \right] \cdot d \vec r$$

With $d \vec r = \vec r'(t) \space dt = \left[\frac{dx}{dt} \hat i + \frac{dy}{dt} \hat j + \frac{dz}{dt} \hat k \right] dt = \left[dx \hat i + dy \hat j + dz \hat k \right]$, we may write:

$$\oint_C \vec F \cdot d \vec r = \oint_C \left[P \hat i + Q \hat j + R \hat k \right] \cdot d \vec r = \oint_C \left[P \hat i + Q \hat j + R \hat k \right] \cdot \left[dx \hat i + dy \hat j + dz \hat k \right]$$

Cleaning up the integral:

$$\oint_C P \space dx + Q \space dy + R \space dz$$

So we conclude:

$$\oint_C \vec F \cdot d \vec r = \oint_C P \space dx + Q \space dy + R \space dz$$

Some food for thought.

8) What is the definition of a simple region?

9) What is the definition of a closed region? What is the definition of a simple region? What is the definition of a conservative vector field?