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o-na [289]
3 years ago
7

Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of

F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = x i − z j + y k S is the part of the sphere x2 + y2 + z2 = 25 in the first octant, with orientation toward the origin
Mathematics
1 answer:
olga_2 [115]3 years ago
5 0

Parameterize S by

\vec s(u,v)=5\cos u\sin v\,\vec\imath+5\sin u\sin v\,\vec\jmath+5\cos v\,\vec k

with 0\le u\le\dfrac\pi2 and 0\le v\le\dfrac\pi2.

Take the normal vector to S to be

\vec s_u\times\vec s_v=-25\cos u\sin^2v\,\vec\imath-25\sin u\sin^2v\,\vec\jmath-25\cos v\sin v\,\vec k

Then the integral of \vec F across S is

\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\int_0^{\pi/2}\int_0^{\pi/2}\vec F(x(u,v),y(u,v),z(u,v))\cdot(\vec s_u\times\vec s_v)\,\mathrm du\,\mathrm dv

\displaystyle=-125\int_0^{\pi/2}\int_0^{\pi/2}(\cos u\sin v\,\vec\imath-\cos v\,\vec\jmath+\sin u\sin v\,\vec k)\cdot(\cos u\sin^2v\,\vec\imath+\sin u\sin^2v\,\vec\jmath+\cos v\sin v\,\vec k)\,\mathrm du\,\mathrm dv

=\displaystyle-125\int_0^{\pi/2}\int_0^{\pi/2}\cos^2u\sin^3v\,\mathrm du\,\mathrm dv

=\displaystyle-125\left(\int_0^{\pi/2}\cos^2u\,\mathrm du\right)\left(\int_0^{\pi/2}\sin^3v\,\mathrm dv\right)=\boxed{-\frac{125\pi}6}

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If x+y=16 and x-y=-2, what is the value of xy?
Fantom [35]

Answer:

Y=7 and

X=9

Step-by-step explanation:

X+Y=16

X=2+Y

2+Y+Y=16

2+2Y=16

2Y=16-2

2Y=14

2Y/2=14/2

Y=7

X =2+Y

So X=9

8 0
9 months ago
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3 years ago
How many 4 digit multiples can not be divided by 5
Advocard [28]

Answer:

1800

Step-by-step explanation:

Let us solve this question by Modular Airthmatic.

Let the lowest 4 digit number be X.

X= 1 (mod 5)

X =2 (mod 5)

X=3 (mod 5)

X =4 (mod 5)

X=1+5 t where t is any integer.

1+5 t=2 (mod 5)

5 t=1 (mod 5 )

5 is the Modular Inverse of modulo 5

t=1 (mod 5)

t =1+5 u : u= any integer.

X becomes 1+ 5 (1+5u )=6+25 u

6+25u =3 (mod 5)

25 u= -3 (mod 5)=2 (mod 5)

25 is Modular Inverse of Modulo 5

u=2 (mod 5)

u =2+5 v : v= any integer.

X becomes 6 +25 (2+5 v)=56+125 v .

56 +125 v=4 (mod 5)

125 v=-52 (mod 5)=3 (mod 5)

125 is Modular Inverse of modulo 5

v = 3 (mod 5)

v =3 +5 w where w is any integer.

For lowest 4 digit number, take

w=200

X=1003 : Lowest 4 digit number.

Take w=201

X=208 : Next number.

For 4 digit largest number, take

w=1995

X= 9978 : Largest number.

No. of Numbers =(1995 - 200 )+1=1800 □Ans.

8 0
3 years ago
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Is 2/4 less than 2/3?
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Yes 2/4 is less than 2/3 :) I hope this helps
5 0
2 years ago
Read 2 more answers
Determine the exact and approximate circumference of the circle.
madam [21]

Answer:

C (circumference) =2×π×r

we've got a 30-60-90 triangle

which means 9=d√3, d=5.19 exact inches

or ≈5.20 inches

d=2r, r=5.20÷2≈2.60 inches

C=2×3.14×2.60≈16.328

3 0
2 years ago
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