Stokes' theorem says that the surface integral of the curl of the vector field F across the surface S is equal to the line integral of F along the boundary of S.
In this case, the boundary is a circle in the x,y-plane with radius 2. (To see this, set z = 0 and rearrange the equation for the paraboloid as x² + y² = 4.)
Then
[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_{\partial S} \vec F \cdot d\vec r[/tex]
where
[tex]\vec r = 2\cos(t) \,\vec\imath + 2\sin(t) \,\vec\jmath[/tex]
parameterizes the circular boundary of S with 0 ≤ t ≤ 2π. Then
[tex]d\vec r = \dfrac{d\vec r}{dt}\,\dt = (-2\sin(t)\,\vec\imath + 2\cos(t)\,\vec\jmath)\,dt[/tex]
and the integral becomes
[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_{\partial S} \vec F \cdot d\vec r[/tex]
[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_0^{2\pi} \vec F(\vec r(t)) \cdot \frac{d\vec r}{dt}\,dt[/tex]
[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s \\= \int_0^{2\pi} ((2\cos(t))^2 \sin(0)\,\vec\imath + (2\sin(t))^2\,\vec\jmath + (2\cos(t))(2\sin(t))\,\vec k) \cdot(-2\sin(t)\,\vec\imath + 2\cos(t)\,\vec\jmath)\,dt[/tex]
[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_0^{2\pi} 8\sin^2(t)\cos(t) \, dt[/tex]
[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \int_0^{2\pi} 8\sin^2(t) \, d(\sin(t))[/tex]
[tex]\displaystyle \iint_S (\nabla\times\vec F)\cdot d\vec s = \frac83 (\sin^3(2\pi)-\sin^3(0)) = \boxed{0}[/tex]
A consumer group is investigating the number of flights.
A gardener bought 5 rabbits.
...
Thats really nice
For the lunch special at a high school cafeteria, students can get either salad or french fries as a side order. The following table shows the number of each side order for the lunch specials purchased on one day, classified by the grade of the student.
The probability that the student that's selected will be in grade 10 given that the student ordered French fries as the side will be 71/248.
From the information given, the number of students that are in grade 10 is 105. Out of all the students, 34 had salad and 71 had French fries. The total students in the school that are French fries was 248.
Therefore, the probability that the student that's selected will be in grade 10 given that the student ordered French fries as the side will be calculated thus:
= Number of students who French fries in grade 10 / Total number of people that had French fries
= 71/248
In conclusion, the correct option is 71/248.
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Marianna is working on improving her communication skills. To be a good listener she should.
Tim drives at an average speed of 80km.
Answer:
and? I dont know Tim.
Explanation:
How does the author unfold the details of the text painter of light.
The author unfolds the details of the "Painter of Light" text when presenting Kinkade's life chronology.
We can arrive at this answer because:
"Painter of Light" is a text with a chronological structure, which presents a series of events that occurred in a timeline.In this text, this kind of structure is used for Kinkade's life chronology, showing the most important events in his life and how that made him relevant to introduce to readers.It is important to point out that textual structure is the way a text is organized to present information to the reader.
More information:
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Answer: The answer is A
Explanation:
1) What is The Full form Of WWW ?
2)What Is The full form of 3 Cs ?
2nd question is little confused .
plz Don't Spam .
WWW is an abbreviation for World Wide Web ...
Corporation, Customer, Competitors (3C's) Model is a business model that focuses on the three key factors needed for success. This model was conceived by the famous business strategist, Kenichi Ohmae.
Answer:
• Full form of WWW :- World Wide Web WWW is an abbreviation for World Wide Web and it is commonly called the WEB. It is a system that is used to store documents and is accessed using URL(Uniform Resource Locator) which are accessible through Internet .• The full form of 3 Cs :- Corporation, Customer, Competitors (3C's) Model is a business model that focuses on the three key factors needed for success. This model was conceived by the famous business strategist, Kenichi Ohmae.Explanation:
Hope this helps you !!Help plz lol………………………….
Answer:
105/330
Explanation:
drama:
25 students
both:
60 students
sports:
140 students
total:
225 students
330-225=105
Given three consecutive integers, if the sum of the first and third integers is increased by 8, the result is 5 less than triple the second integer. Find the three consecutive integers. *
Answer:
The three consecutive integers are 12, 13 & 14
Explanation:
Let the three consecutive integers be x, x + 1, x + 2
According to the question,
If sum of the first and third integers is increased by 8, the result is 5 less than triple the second integer.
[tex] \therefore [/tex] x + x + 2 + 8 = (x + 1) × 3 - 5
2x + 10 = 3x + 3 - 5
2x + 10 = 3x - 2
10 + 2 = 3x - 2x
x = 12
So, The three consecutive integers are 12, 13 & 14
Rami is graduating high school in the spring. He is interested in becoming a electrician. What are his options for learning this trade? select all that apply.
To be a licensed electrician requires deep knowledge of the field. Rami's options for learning this trade are;
He should applied degree at a two year college or He can study for a bachelor's degree at a four year collegeApprenticeship and trade schoolA licensed electrician job is not easy to obtain. It requires studying, years of being an apprentice or attending school, and years of work experience.
To be a licensed electrician, these steps should be followed:
Earn a high school diploma or Its equivalent. Try and attend a trade or vocational-technical school. Sign up as an electrician apprentice. Get licensed or certified.See full question blow
Rami is graduating high school in the spring. He is interested in becoming a electrician. What are his option learning this trade? Select all that apply.
applied degree at a two year college bachelor's degree at a four year college vocational training in the military apprenticeship and trade school going directly into the workplaceLearn more from
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Answer: 1) applied degree at a two year college
3) Vocational training in the military
4) apprenticeship and trade school
Explanation: did on edge :)
Imagine that you are a writer for a leading newspaper company in the philippines brainly.
If I was a writer, the implementation of a 24-hour curfew for minors in time of pandemic is vital in order to keep them safe.
This is an incomplete question. The question is about the implementation of a 24-hour curfew for minors in time of pandemic. A pandemic is an epidemic of an infectious disease which has spread over large regions and has affected several individuals.
During a pandemic, there's an outbreak of diseases that can be transmitted from person-to-person contact. Therefore, it's important for people to be safe.
To address this issue, there should be an implementation of a 24-hour curfew for minors in the time of pandemics. This is vital to keep them safe.
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4 t-shirts and a hat costs £31. 00 2 t-shirts and a hat costs £17. 0. How much does a t-shirt cost? how much does a hat cost?.
Answer:
T-Shirt Cost = £7
Hat Cost = £3
Explanation:
Important Information:
4 t-shirts and a hat costs £31.00
2 t-shirts and a hat costs £17.00.
Here we assume t-shirt be x and hat be y.
Explanation/Solution:
Based on the above information, the calculation is as follows:
4x + y = 31
2x + y = 17
Now subtract equation 2 from 1
2x = 14
x = 7
So y is
4(7) + y = 31
28 + y = 31
y = 3
Therefore we can conclude that The t-shirt cost £7 and Hat cost £3.
