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]
How did buddhism influence east asian arts and crafts.
Adopting the Buddhist way of paying homage through art, Taoist art began being created and China developed its architectural culture.
Which of the following most directly contributed to the change over time depicted on the two maps?.
Locations are often shown on a map. The option that directly contributed to the change over time depicted on the two maps is the The building of canals and roads.
In 1800, U.S. citizens and immigrants to the country often traveled majorly by horseback or on the rivers.
As time went by, crude roads and canals were made. It was later on that railroads was made which was used to move people and goods with greater efficiency.Conclusively, A map is shows the visual representation of a location and the various part that is found in that specific location.
See full question below
Which of the following most directly
contributed to the change over time depicted on
the two maps?
(A) The construction of a national highway
system
(B) The establishment of different time zones
across the country
(C) The building of canals and roads
(D) The use of diesel engines
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Which notations represent atoms that have the same number of protons.
Answer:
ITS ABOUT DRIVE ITS ABOUT POWER WE STAY HUNGRY WE DEVOUR PUT IN THE WORK PUT IN THE HOURS AND TAKE WHATS OURS
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.
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Answer: The answer is A
Explanation:
Who intended to write an epic poem 12 books long? Question 3 options: Francis Drake Sir Walter Raleigh Edmund Spenser William Shakespeare.
Answer:
emund spencer
Explanation:
k12 test got it right have a great day xD
Excessive smoking, substance abuse, and sleep disorders are all examples of ________ symptoms of stress.
Excessive smoking, substance abuse, and sleep disorders are considered to be behavioral symptoms of stress.
It should be noted that behavior-related stress symptoms include absence and turnover, changes in productivity, changes in eating habits, increased consumption of alcohol, and smoking.
In a case where an individual who doesn't usually drink alcohol or smokes before starts drinking alcohol, it's known as behavioral symptoms of stress. In such a situation, stress leads to a change of behavior.
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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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Which of the following distinguishes prophase 1 of meiosis from prophase of mitosis?.
Based on a scientific study, the difference between prophase 1 of meiosis from prophase of mitosis is "homologous chromosomes pair up."
prophase 1 of meiosis vs. prophase of mitosisProphase 1 of meiosis is the first stage of meiosis which involves recombination and alleles crossing over genetic material to form non-identical haploid chromatids.
In Contrast, the prophase of mitosis is also the first stage of mitosis which splits the duplicated genetic fabric present in the nucleus of a parent cell to form new identical daughter cells.
However, one of the significant differences between prophase 1 of meiosis and prophase of mitosis in appearance is " homologous chromosomes pair up."
Hence, in this case, it is concluded that the correct answer is "homologous chromosomes pair up."
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A family has two cats named gordo and flaco. Gordo weighs 15 pounds and flaco weighs 8 pounds. A cat’s weight is classified as unhealthy if the weight is located in the top 5% or bottom 5% of all cat weights. The distribution of cat weights is approximately normal with mean 9. 5 pounds and standard deviation 1. 5 pounds. Which of the following is the best description of gordos and flacos weights?.
Would you recommend dancing as an activity done at home.
Which description from "sugihara" best indicates that several hours have passed?.
Answer:
its A
Explanation:
hope this helps, pls lmk if its wrong
Help plz lol………………………….
Answer:
105/330
Explanation:
drama:
25 students
both:
60 students
sports:
140 students
total:
225 students
330-225=105
How many grams of iron (density = 7. 87 g/ml ) would occupy the same volume as 96. 4 g of aluminum (density = 2. 70 g/ml)?.
Explanation:
density = mass / volume
96.4 g of aluminum / volume = 2.7
therefore,
96.4 = 2.7 × volume
volume = 96.4 / 2.7 = 35.7037037... ml
back to iron :
7.87 = mass / 35.7037037...
mass = 7.87 × 35.7037037... = 280.9881481... g
it would be 280.9881481... g of iron in the same volume.
In casabianca the boy's defining character trait is his (points 5).
The character trait that defines the boy in Casabianca is: his loyalty.
Casabianca is the name of the young boy who is one of the major character sin "Casabianca".
In Casabianca, the boy, out of loyalty and having a strong sense of duty, had to put up bravery to remain at his post on a French ship that caught fire during the attack of the British.
In spite of the obvious dangers, he remained at his post while waiting for orders from the captain (his dad).
Therefore, the character trait that defines the boy in Casabianca is: his loyalty.
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List at least two facts you learned from your research that you could use when writing the opening statement. Remember to include information that is relevant to the main problem you are investigating.
Two facts that I learned during my research were narrowing down the research topic and setting appropriate goals.
When doing research, several aspects must be taken into account so that the research has good results and produces new knowledge. Here are two tips for proper research:
1. Limit the subject of the investigation: When conducting an investigation it is necessary to limit the subject so that it is easier to achieve an effective result. For example, time limits, a specific place, and more special aspects should be set to make the investigation more specialized.
2. Establish adequate objectives: When conducting an investigation it is necessary to establish objectives that can be achieved in the investigation. Highly ambitious goals can make research and information search difficult.
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Answer:
-The cost of college has risen faster than inflation for many years.
-The high price of student services and administration are two reasons for the rapid increase in cost.
Explanation:
good luck guys
A ratio estimates the risk associated with investing in a firm.
A ratio which estimates the risk associated with investing in a business firm is called: solvency ratio.
Solvency ratio can be defined as a key metric that is typically used to measure the ability of a business firm to meet its long-term debt obligations.
Basically, a solvency ratio measures the financial position of a business firm and the extent to which its assets cover long-term debt obligations (commitments), especially for future payments and the liabilities.
In conclusion, a ratio which estimates the risk associated with investing in a business firm is called solvency ratio.
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A wave with short wavelength will have ____ frequency and ___ energy.
According to the electron-cloud model of the atom an orbital is a.
Answer:
orbital is a axis where electron revolvs
An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed with mean equal to 800 hours and a standard deviations of 40 hours. Find the probability that, a random sample of 16 bulbs will have an average life of i). Less that 775 hours. Ii) between 775 and 825 hours.
Answer:
ion know
Explanation:
Between ant and grasshopper which would you choose to be why.
Answer:
ant
Explanation:
way more interesting, can carry objects way heavier than them and if they were to become human size they would pretty much rule the earth whereas a grasshopper is annoyinggggfg and if they were the size of humans we'd probably die of blood loss in the ears.
When it comes to the choice between an ant and a grasshopper, my choice would be an ant because:
Ants are really strong. Ants are farmers. Ants are very resilient as a species.Ants are known to be really strong such that they can carry things that are 50 times their body weight which allows them to carry a lot of food to their colonies.
Ants are also farmers in that they have been known to protect some other insects so that they can keep getting agricultural produce like honeydew.
Ants are also quite resilient in that they have been around for as long as dinosaurs which means they are some of the oldest species in the world.
In conclusion, ants are preferably to grasshoppers in my opinion.
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Write an informative essay explaining what has caused the english spoken today to be different.
Answer:
fortnit
Explanation:rr
What is the sum of the polynomials (8x2-9y2-4x)+(x2-3y2-7x).
The answer to your question is: D.) 9x²-11x-12y²
answer:
9x-2-12y-2-11x
explanation:
Remove the parentheses: 8x-2-9y-2-4x-+x-2-3y-2-7x
Combine like terms: 9x-2-12y-2-11x
Answer: 9x-2-12y-2-11x
Which functions have a vertex with a x-value of 0? select three options.
87 AND (86/) BE BACK
Which set of equations is enough information to prove that lines c and d are parallel lines cut by transversal p? m∠1 = 81° and m∠2 = 99° m∠3 = 99° and m∠4 = 99° m∠2 = 99° and m∠4 = 99° m∠4 = 81° and m∠1 = 81°.
The set of equation that is enough information to prove that lines c and d are parallel lines cut by transversal p is m∠2 = 99° and m∠4 = 99°. The c.
What are transverse lines?
A transversal line is a line that crosses two or more other lines at different locations. Line l meets a and b in the image below at two different locations, P and Q. Line l is the transversal line as a result.
When a transversal cuts two lines so that their subsequent interior angles are supplementary, the lines are said to be parallel. Two lines are parallel to one another if they are both parallel to the same line. When the marked consecutive internal angles are supplementary, lines m and n are parallel.
Therefore, the correct option is c, m∠2 = 99° and m∠4 = 99°.
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The question is incomplete. Your most probably complete question is given below: the image is added.
The set of equation that is enough information to prove that lines c and d are parallel lines cut by transversal p is m∠2 = 99° and m∠4 = 99°. The c.
What are transverse lines?A transversal line is a line that crosses two or more other lines at different locations. Line l meets a and b in the image below at two different locations, P and Q. Line l is the transversal line as a result.
When a transversal cuts two lines so that their subsequent interior angles are supplementary, the lines are said to be parallel. Two lines are parallel to one another if they are both parallel to the same line. When the marked consecutive internal angles are supplementary, lines m and n are parallel.
Therefore, the correct option is c, m∠2 = 99° and m∠4 = 99°.The set of equation that is enough information to prove that lines c and d are parallel lines cut by transversal p is m∠2 = 99° and m∠4 = 99°. The c.
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A package of nuts contains 3 servings, and each serving contains 150 calories. If you eat the entire package, how many calories did you consume?.
Answer:
450 calories 150 x 3 equals 450
How does the first amendment protect religious freedom for american citizens? choose exactly two answers that are correct.
Answer:
It lets people practice their faith in their own way and
It prevents the government from establishing a national religion.
Explanation:
A gardener bought 5 rabbits.
...
Thats really nice
According to the table, what was the hiker’s average speed for the entire hike? 2. 0 km/h 5. 7 km/h 6. 7 km/h 10 km/h.
Answer:
6.7 km/h
Explanation:
A company upgraded production machinery to capture data about the fit tolerance for assembled parts of each unit produced. Before the upgrade, inspectors created the same data by sampling 10 percent of units produced. The upgrade represents an increase in.
The upgrade in production machinery to capture data about the fit tolerance for assembled parts of each unit produced represents an increase in: data volume.
Big data can be defined as huge collections of data that are typically difficult for users to process, analyze, and manage through the use of conventional data tools.
Basically, big data comprises five (5) "Vs" and these are:
VelocityVeracityValueVarietyVolumeData volume refers to the amount of data that are stored in a database by an organization for analysis and processing. Thus, data volume represents an organization's data size.
In this scenario, the upgrade in production machinery to capture data about the fit tolerance for assembled parts of each unit produced represents an increase in data volume.
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How much force is needed to stop a 100-kg football player if she decelerates at 20 m/s2? 5 N 20 N 200 N 2000 N.
2000 N. is needed to stop a football player