Use Stoke's theorem to evaluate ?

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In summary: Secondly, how do we use it to evaluate this particular integral?In summary, a student is seeking help with a problem in calculus II, specifically using Stokes' theorem to evaluate an integral from their textbook. They mention their professor giving them a preview of what they will learn in calculus III and asking them to work on problems together. The student is asking for help understanding the problem and wanting to make a good impression on their professor.
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CalleighMay
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Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help!

I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is our last week of class after our final exam so my professor is taking this time to give us a preview of what we will be learning in the fall semester in Calc III (since this is the same professor). Every Tuesday class our professor gives us a few problems from future sections and asks us to "see what we can come up with" and to work together to find solutions. The following Tuesday he asks us to discuss the problems as a class, seeing which ones of us know our stuff =P

Basically, i want to ask you guys what you think about these problems as i do them along before i have my discussion. I really want to make a lasting impression on my professor by "knowing my stuff" -to show him i can do it! All's i need is a little help! Would you guys mind giving me some help?

We are using the textbook Calculus 8th edition by Larson, Hostetler and Edwards and the problems come from the book.

The problem is on pg 1133 in chapter 15.8 in the text, number 14. It reads:

Use Stoke's theorem to evaluate Integral (with C at bottom) of F (with a dot) dr
It states that in each case, C is oriented counterclockwise as viewed from above.
For this specific problem is gives,
F(x,y,z)=4xzi + yj + 4xyk
and
S: z= 9-x^2-y^2
and z>=0

Again, i literally haven't a clue where to go with these =/ I looked at the soln's to the other problems in this set but still haven't a clue where to go! =/
Any help at all would be greatly appreciated. Thanks guys ;)
 
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Well, firstly, what is Stokes' theorem?
 

1. What is Stoke's theorem?

Stoke's theorem is a mathematical theorem that relates the line integral of a vector field over a closed curve to the surface integral of the curl of the vector field over the surface enclosed by the curve.

2. When is Stoke's theorem used?

Stoke's theorem is used in vector calculus to evaluate integrals of vector fields over closed curves or surfaces. It is also used in many applications, such as fluid mechanics and electromagnetism.

3. What is the formula for Stoke's theorem?

The formula for Stoke's theorem is: ∫F⋅ds = ∮(curl F)⋅dS, where F is the vector field, ds is the line element, and dS is the surface element.

4. How is Stoke's theorem related to the Fundamental Theorem of Calculus?

Stoke's theorem is a higher-dimensional extension of the Fundamental Theorem of Calculus. Both theorems relate integrals of a function over a curve or surface to the derivatives of the function.

5. What are some real-world applications of Stoke's theorem?

Stoke's theorem has many practical applications in fields such as physics, engineering, and fluid mechanics. For example, it is used to calculate the circulation of fluid flow around a closed curve or the magnetic flux through a closed surface.

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