1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
puteri [66]
3 years ago
7

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = 2yi + xzj + (x

+ y)k, C is the curve of intersection of the plane z = y + 6 and the cylinder x2 + y2 = 1.
Mathematics
1 answer:
Lady_Fox [76]3 years ago
4 0

By Stokes' theorem, the line integral of \vec F over C is given by the surface integral of the curl of \vec F over S, where S is the region of intersection of the plane z=y+6 and the cylinder x^2+y^2=1 with S having positive/upward orientation.

Parameterize S by

\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(u\sin v+6)\,\vec k

with 0\le u\le1 and 0\le v\le2\pi.

Take the normal vector to S to be

\vec s_u\times\vec s_v=-u\,\vec\jmath+u\,\vec k

The curl of \vec F is

\nabla\times\vec F(x,y,z)=(1-x)\,\vec\imath-\vec\jmath+(z-2)\,\vec k

Then the line integral is equivalent to

\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S

=\displaystyle\int_0^{2\pi}\int_0^1\bigg((1-u\cos v)\,\vec\imath-\vec\jmath+(u\sin v+4)\,\vec k\bigg)\cdot\bigg(-u\,\vec\jmath+u\,\vec k\bigg)\,\mathrm du\,\mathrm dv

=\displaystyle\int_0^{2\pi}\int_0^1(5u+u^2\sin v)\,\mathrm du\,\mathrm dv=\boxed{5\pi}

You might be interested in
A bus travels 10km in 15 min if the speed of the bus remains the same how far would the bus travel in 3hrs?
motikmotik

Answer:

120km

Step-by-step explanation:

10km in 15 min

3hrs = 3 x 60 min = 180 min

180min divided by 15min = 12

12 x 10km = 120 km

3 0
3 years ago
Read 2 more answers
Can somebody answer this question
Trava [24]
I can’t really see please text it
7 0
3 years ago
Read 2 more answers
PLEASE HELP GIVING BRAINLIEST
77julia77 [94]

Answer:

The answer is A)

Step-by-step explanation:

When the plane length increases, the force required gets reduced while the plane height stays the same.

<em>Hope this helps!</em>

3 0
2 years ago
- 3x + 5y = -2<br> 2x - 2y = 1<br> Solve by elem inaction
Lyrx [107]

Answer:

x=1/4 , y=-1/4. You can confirm this by substituting the numbers in.

Step-by-step explanation:

4 0
3 years ago
Write three fractions equivalent to 27/30. show work
Kipish [7]
54/60 108/120 216/240 double both numbers each time
4 0
3 years ago
Other questions:
  • The sum of a number and 7 divided by -3 is 2
    7·2 answers
  • Convert 3.2 repenting decimal into a fraction
    9·1 answer
  • (fog)(x) and (gof)(x)<br> f(x)= 8/1-5x<br> g(x)= 1/x<br><br> Then find the domain for each
    7·1 answer
  • Please helpppppppppp
    7·1 answer
  • A person is working at a constant rate. If he produces 15 details in 8 hours, how long does he need to work to produce 18 detail
    11·1 answer
  • Help plz ? Mark brainliest?
    5·1 answer
  • Consider the figure shown. what is the measure of the given angles?
    14·1 answer
  • Prove that:<br><br>cos20°cos40°cos80°=1/8​
    12·2 answers
  • Suppose that the length of a certain rectangle is two centimeters more than three times its width. If the area of the rectangle
    7·1 answer
  • If there are 25 animals in total, how many dogs are at the kennel?
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!