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Ugo [173]
4 years ago
10

Calculate the circulation, F · dr, C in two ways, directly and using Stokes' Theorem. F = y i + z j + xk and C is the boundary o

f S, the paraboloid z = 4 − (x2 + y2), z ≥ 0 oriented upward. (Hint: Use polar coordinates.) F · dr C =
Mathematics
1 answer:
otez555 [7]4 years ago
3 0

C, the boundary of S, is a circle in the x,y plane centered at the origin and with radius 2, hence we can parameterize it by

\vec r(t)=2\cos t\,\vec\imath+2\sin t\,\vec\jmath

with 0\le t\le2\pi. Then the line integral is

\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\int_0^{2\pi}(2\sin t\,\vec\imath+2\cos t\,\vec k)\cdot(-2\sin t\,\vec\imath+2\cos t\,\vec\jmath)\,\mathrm dt

=\displaystyle\int_0^{2\pi}-4\sin^2t\,\mathrm dt

=\displaystyle-2\int_0^{2\pi}(1-\cos2t)\,\mathrm dt=\boxed{-4\pi}

By Stokes' theorem, the line integral of \vec F along C is equal to the surface integral of the curl of \vec F across S:

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

Parameterize S by

\vec s(u,v)=u\cos v\,\vec\imath+u\sin v\,\vec\jmath+(4-u^2)\,\vec k

with 0\le u\le2 and 0\le v\le2\pi. Take the normal vector to S to be

\vec s_u\times\vec s_v=2u^2\cos v\,\vec\imath+2u^2\sin v\,\vec\jmath+u\,\vec k

The curl is

\nabla\times\vec F=-\vec\imath-\vec\jmath-\vec k

Then the surface integral is

\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S=\int_0^{2\pi}\int_0^2(-\vec\imath-\vec\jmath-\vec k)\cdot(2u^2\cos v\,\vec\imath+2u^2\sin v\,\vec\jmath+u\,\vec k)\,\mathrm du\,\mathrm dv

=\displaystyle-\int_0^{2\pi}\int_0^2(2u^2\cos v+2u^2\sin v+u)\,\mathrm du\,\mathrm dv=\boxed{-4\pi}

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Word problem plz help pt.1​
shusha [124]

equation csn be represented as

money left = 500 - (n-1)×4.65

so we can see in first week money left

= 500 - (1-1)×4.65= 500-0=500

which is true because first week chaya got gift card and does not spend any money.

now lets assume that on week n money left is 327.95

so by equation

500 -(n-1)×4.65 = 327.95

500- 327.95 =( n-1)× 4.65

n -1 = 172.05/4.65=37

n = 37+1 = 38

in 38th week money left would be 327.95

3 0
3 years ago
As an estimation we are told 5 miles is 8 km. Convert 17.8 km to miles.
kherson [118]

Answer:

17.8km = 11.125 miles

Step-by-step explanation:

From 5 miles = 8km, we can form the ratio km:miles as 8 : 5.

We want to put the ratio in the form 1 : n, which gives us 1 : 0.625 (we divide both 8 and 5 by 8 to get this form).

Now, using the ratio km:miles as 1:0.625, we have to multiply both 1 and 0.625 by 17.8 to gain the ratio 17.8 : n, therefore giving us the ratio 17.8 : 11.125.

Hence, 17.8km = 11.125 miles

6 0
3 years ago
Read 2 more answers
Please help me answer this.
adoni [48]

Answer:

See explanation

Step-by-step explanation:

        Statements                         Reasons

1. \overline{AD}\parallel \overline{BC}                                        Given

2. \angle ADB\cong \angle CBD      As alternate interior angles when parallel lines AD and BC intersect by ltransversal BD

3. \overline{AD}\cong \overline{BC}                                     Given

4. \overline{BD}\cong \overline{DB}                     Reflexive property

5. \triangle ADB\cong \triangle CBD                  SAS postulate

6.  \angle ABD\cong \angle CDB              Corresponding parts of congruent triangles are congruent

7. \overline{AB}\parallel \overline{CD}            Inverse alternate interior angles theorem

8 0
3 years ago
5/8 to the power of the 2
MatroZZZ [7]
<span>2 <span>14/<span>25 as a mixed number decimal form is 2.56 and exact form is 64/25



hope i helped best of luck :)  </span></span></span>
<span />
8 0
3 years ago
For the data set below identify the interquartile range of the data.
Makovka662 [10]

To find the range of a set, subtract the largest number and the smallest number. Those numbers, in this set, are 63 and 34.

63-34=29

So the range is 29

---

sorry if I didn't help

6 0
3 years ago
Read 2 more answers
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