Firstly, I know how to do most of it but Im feeling lazy. So:

Id like a full, organized and clearly written solution please.

If its possible to get two solutions from different two people, please do that and charge me for it.

Id like at least 90% score on this assignment.

So, send it to the best. You got three days

Thank you.

Rubric:

This exam will be a little different from traditional exams. The first three problems in this exam are open-ended prompts. Rather than being asked to perform a specific computation like on the midterm exam, you will instead be given an open-ended prompt that can be responded to in many different ways. You will find the prompt in a box on each problem.

The purpose of this open-ended exam is to give you the opportunity to show off your mastery of the material in your own way. You aren’t comfortable with Jacobians? Great, you don’t have to talk about them. You really like coming up with analogies to describe complicated ideas? Great! Let’s hear them!

Each prompt will give you suggestions for responding. You should take these suggestions into account in your responses, but these are just suggestions! Do not simply respond to the bullet-point suggestions. You also do not necessarily have to respond to each suggestion; if there is another aspect of the problem that you would like to analyze, you are free to do so! You are expected to take the time to formulate your own unique responses to the problems. You should consider each problem as really telling you “prove that you understand the following concept”.

Your submissions are graded on both correctness and effort. Remember: this is a final exam, which means your best effort is expected. Your responses will likely involve both computations, pictures, and narration. Make sure to write in full sentences when necessary! Of course, you are not required to always write formally. You will not be judged for grammar mistakes, but you should make an effort to have your responses be legible. I am more interested in the content of your responses than its appearance.

The following qualities of your responses will be considered when grading the responses to Problems 1, 2, and 3.

• (10 Points) Comprehensiveness.

1. Full points are awarded to a response that analyzes multiple aspects of the prompt, making use of multiple relevant techniques that were covered this term. The student shows knowledge of most of the relevant concepts that could be used to respond to the prompt. More than one of these concepts are analyzed in-depth, suggesting that the student deeply understands the material.

2. 5-9 points are awarded to a response that analyzes the prompt from one or two different aspects with varying levels of detail, or only makes a shallow analysis of the prompt from multiple angles.

3. 0-4 points are awarded to a response that does not analyze the prompt in sufficient detail. This response may only consider a single relevant concept and does so without much detail.

• (10 Points) Correctness.

1. Full points are awarded to a response where a majority of the arguments and computations are correct.

2. 5-9 points are awarded to a response that makes only one or two major logical or computational errors.

3. 0-4 points are awarded when the response contains multiple errors.

• (10 Points) Creativity, Effort, and Organization.

1. Full points are awarded to a response that suggests creative thinking and effort. The response does not simply restate sentences from the textbook (except where necessary, such as in the statement of a theorem), but reveals the student’s unique voice in writing. The student goes out of their way to make their arguments clear using additional diagrams, commentary, or analogies. The response is also well-organized and easy to read.

2. 5-9 points are awarded to a response that is less organized and shows less effort than described in the full-point category. The response may be more cumbersome to read and does not suggest that the student shows a full understanding of the concepts used in their response. The response does not go far beyond simply responding to the suggestions given in each prompt.

3. 0-4 points are awarded to a response that is not well-organized and only responds to the suggestions with minimal effort.

If you have any questions about any part of the exam, please do not hesitate to email me. I will be on high-alert for emails during our final exam week!

Vector Field Exploration!

This is your opportunity to show how you could apply the techniques used in this class to analyze vector fields in applications.

Here are some ideas to get you started:

• How could you visualize the geometry of this vector field? What kind of motion does it describe?

• What desirable properties does this vector field have (if any?).

• Under what circumstances do our favorite theorems apply to this vector field?

WARNING: do not simply respond to the bullet points! These bullet points are meant to help you come up with ideas! Remember, your response is graded on correctness and effort: writing a simple response to each of the bullet points will be considered minimal effort.

Geometry Exploration!

This is your opportunity to show off your understanding of the domains of integration for line and surface integrals.

2. Discuss the role that the geometry of curves and surfaces plays in vector calculus. In particular, how do we use calculus to describe (parametrize) curves and surfaces for the integration of multivariable functions? How does this interplay of calculus and geometry benefit us in our investigation of scalar functions and vector fields?

Here are some ideas to get you started:

• What are some common parametrizations of curves? What are some common parametrizations of surfaces? How do you know that these parametrizations are correct, and where do they come from?

• Once you have a parametrization of a curve, what do you do with it to compute a line integral?

• Once you have a parametrization of a surface, what do you do with it to compute a surface integral?

• What about going backwards: if you have a parametrization of a curve or surface, how do you determine the shape that the function describes?

WARNING: do not simply respond to the bullet points! These bullet points are meant to help you come up with ideas! Remember, your response is graded on correctness and effort: writing a simple response to each of the bullet points will be considered minimal effort.

Fundamental Theorem of Calculus Exploration!

The purpose of this problem is for you to show you understand both Stokes’ theorem and the divergence theorem, including when these theorems can be used and how they can be used.

• What is the fundamental theorem of calculus?

• Compute the work done by F~ over your own choice of curves C using Stokes’ theorem, explaining every step of the way.

• Compute the flux of F~ through your own choice of surfaces S using the divergence theorem, explaining every step of the way.

• Are Stokes’ theorem and the divergence theorem necessarily “shortcuts”? What would it look like if you computed the work or flux directly using the original definitions without the theorems?

WARNING: do not simply respond to the bullet points! These bullet points are meant to help you come up with ideas! Remember, your response is graded on correctness and effort: writing a simple response to each of the bullet points will be considered minimal effort.

Note: You do not need to go through the entire computation of the integrals, as long as you set up each integral with everything simplified as much as possible.

4. (10 Points) Tying Up Loose Ends...

Use Problem 3 from the Midterm Exam to show that the flow curves of a conservative vector field are never closed curves.

(Hint: To prove this, suppose that a flow curve of a conservative vector field actually could be a closed loop. What does Problem 3 from the Midterm Exam say about this curve? This should contradict something that another theorem says about this curve... Once you identify and describe the contradiction, your proof is done.)

Id like a full, organized and clearly written solution please.

If its possible to get two solutions from different two people, please do that and charge me for it.

Id like at least 90% score on this assignment.

So, send it to the best. You got three days

Thank you.

Rubric:

This exam will be a little different from traditional exams. The first three problems in this exam are open-ended prompts. Rather than being asked to perform a specific computation like on the midterm exam, you will instead be given an open-ended prompt that can be responded to in many different ways. You will find the prompt in a box on each problem.

The purpose of this open-ended exam is to give you the opportunity to show off your mastery of the material in your own way. You aren’t comfortable with Jacobians? Great, you don’t have to talk about them. You really like coming up with analogies to describe complicated ideas? Great! Let’s hear them!

Each prompt will give you suggestions for responding. You should take these suggestions into account in your responses, but these are just suggestions! Do not simply respond to the bullet-point suggestions. You also do not necessarily have to respond to each suggestion; if there is another aspect of the problem that you would like to analyze, you are free to do so! You are expected to take the time to formulate your own unique responses to the problems. You should consider each problem as really telling you “prove that you understand the following concept”.

Your submissions are graded on both correctness and effort. Remember: this is a final exam, which means your best effort is expected. Your responses will likely involve both computations, pictures, and narration. Make sure to write in full sentences when necessary! Of course, you are not required to always write formally. You will not be judged for grammar mistakes, but you should make an effort to have your responses be legible. I am more interested in the content of your responses than its appearance.

The following qualities of your responses will be considered when grading the responses to Problems 1, 2, and 3.

• (10 Points) Comprehensiveness.

1. Full points are awarded to a response that analyzes multiple aspects of the prompt, making use of multiple relevant techniques that were covered this term. The student shows knowledge of most of the relevant concepts that could be used to respond to the prompt. More than one of these concepts are analyzed in-depth, suggesting that the student deeply understands the material.

2. 5-9 points are awarded to a response that analyzes the prompt from one or two different aspects with varying levels of detail, or only makes a shallow analysis of the prompt from multiple angles.

3. 0-4 points are awarded to a response that does not analyze the prompt in sufficient detail. This response may only consider a single relevant concept and does so without much detail.

• (10 Points) Correctness.

1. Full points are awarded to a response where a majority of the arguments and computations are correct.

2. 5-9 points are awarded to a response that makes only one or two major logical or computational errors.

3. 0-4 points are awarded when the response contains multiple errors.

• (10 Points) Creativity, Effort, and Organization.

1. Full points are awarded to a response that suggests creative thinking and effort. The response does not simply restate sentences from the textbook (except where necessary, such as in the statement of a theorem), but reveals the student’s unique voice in writing. The student goes out of their way to make their arguments clear using additional diagrams, commentary, or analogies. The response is also well-organized and easy to read.

2. 5-9 points are awarded to a response that is less organized and shows less effort than described in the full-point category. The response may be more cumbersome to read and does not suggest that the student shows a full understanding of the concepts used in their response. The response does not go far beyond simply responding to the suggestions given in each prompt.

3. 0-4 points are awarded to a response that is not well-organized and only responds to the suggestions with minimal effort.

If you have any questions about any part of the exam, please do not hesitate to email me. I will be on high-alert for emails during our final exam week!

Vector Field Exploration!

This is your opportunity to show how you could apply the techniques used in this class to analyze vector fields in applications.

Here are some ideas to get you started:

• How could you visualize the geometry of this vector field? What kind of motion does it describe?

• What desirable properties does this vector field have (if any?).

• Under what circumstances do our favorite theorems apply to this vector field?

WARNING: do not simply respond to the bullet points! These bullet points are meant to help you come up with ideas! Remember, your response is graded on correctness and effort: writing a simple response to each of the bullet points will be considered minimal effort.

Geometry Exploration!

This is your opportunity to show off your understanding of the domains of integration for line and surface integrals.

2. Discuss the role that the geometry of curves and surfaces plays in vector calculus. In particular, how do we use calculus to describe (parametrize) curves and surfaces for the integration of multivariable functions? How does this interplay of calculus and geometry benefit us in our investigation of scalar functions and vector fields?

Here are some ideas to get you started:

• What are some common parametrizations of curves? What are some common parametrizations of surfaces? How do you know that these parametrizations are correct, and where do they come from?

• Once you have a parametrization of a curve, what do you do with it to compute a line integral?

• Once you have a parametrization of a surface, what do you do with it to compute a surface integral?

• What about going backwards: if you have a parametrization of a curve or surface, how do you determine the shape that the function describes?

WARNING: do not simply respond to the bullet points! These bullet points are meant to help you come up with ideas! Remember, your response is graded on correctness and effort: writing a simple response to each of the bullet points will be considered minimal effort.

Fundamental Theorem of Calculus Exploration!

The purpose of this problem is for you to show you understand both Stokes’ theorem and the divergence theorem, including when these theorems can be used and how they can be used.

• What is the fundamental theorem of calculus?

• Compute the work done by F~ over your own choice of curves C using Stokes’ theorem, explaining every step of the way.

• Compute the flux of F~ through your own choice of surfaces S using the divergence theorem, explaining every step of the way.

• Are Stokes’ theorem and the divergence theorem necessarily “shortcuts”? What would it look like if you computed the work or flux directly using the original definitions without the theorems?

WARNING: do not simply respond to the bullet points! These bullet points are meant to help you come up with ideas! Remember, your response is graded on correctness and effort: writing a simple response to each of the bullet points will be considered minimal effort.

Note: You do not need to go through the entire computation of the integrals, as long as you set up each integral with everything simplified as much as possible.

4. (10 Points) Tying Up Loose Ends...

Use Problem 3 from the Midterm Exam to show that the flow curves of a conservative vector field are never closed curves.

(Hint: To prove this, suppose that a flow curve of a conservative vector field actually could be a closed loop. What does Problem 3 from the Midterm Exam say about this curve? This should contradict something that another theorem says about this curve... Once you identify and describe the contradiction, your proof is done.)

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