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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}

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The measure of an angle is 8 degrees less than three times the measure of the angle's supplement. Find the measure of the angle.
Liula [17]

Answer:

262

Step-by-step explanation:

3(90) - 8 = x

270-8 = x

262 = x

4 0
3 years ago
#20 I’m not sure how to solve it. And the answer should be able to grid in like on the SAT
r-ruslan [8.4K]

Answer:

1

Step-by-step explanation:

i (a + bi) = 1

ai + b(-1) = 1

ai − b = 1

Match the coefficients.

ai = 0

-b = 1

Therefore, a = 0 and b = -1.

So a − b = 0 − (-1) = 1.

4 0
3 years ago
The volume of a sphere is two-thirds of a cylinder. What is the volume of a cylinder that has the same diameter as a sphere with
sattari [20]

Answer:

V_2= \frac{28}{3} u^3

Step-by-step explanation:

Given

Represent the volume of the cylinder with V1 and the volume of the sphere with V2

So, from the first statement: we have:

V_2 =\frac{2}{3}V_1

and

V_1 = 14u^3

To solve for V_2, we simply substitute 14u^3 for V_1 in V_2 =\frac{2}{3}V_1

V_2 =\frac{2}{3}V_1

V_2= \frac{2}{3} * 14u^3

V_2= \frac{2* 14}{3} u^3

V_2= \frac{28}{3} u^3

Hence, the volume of the sphere is \frac{28}{3} u^3

5 0
2 years ago
The graphs below have the same shape. What is the equation of the blue<br>graph?<br>​
faltersainse [42]
The equation of the blue graph is A.
4 0
3 years ago
Read 2 more answers
I WILL GIVE BRAINLIEST....PLEASE HELP..........
REY [17]

Answer:  The correct table which represents relationship B is,

Time (min) 1 3 8 9

Temperature (°C) 54.2 59.0 71.0 73.4

Step-by-step explanation:

Since, It is given that, table represents Relationship A.

Time (min) 1 3 8 9

Temperature (°C) 53.6 58.4 70.4 72.8

And, According to the question,  Relationship B has a greater rate than Relationship A.

Therefore, for a particular time Relationship B must be given the greater value than relationship A.

Thus In 1 minute, The temperature by B > 53.6°

In 3 minutes, The temperature by relationship B> 58.4°

In 8 minutes, The temperature by relationship B> 70.4°

In 9 minutes, The temperature by relationship B> 72.8°

But when we apply this we found only Fourth table is following the above conditions.

Therefore, The table,

Time (min) 1 3 8 9

Temperature (°C) 54.2 59.0 71.0 73.4

represented by fourth option will be the correct table for relationship B.




3 0
3 years ago
Read 2 more answers
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