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
My name is Ann [436]
3 years ago
15

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 = 49 in the first octant, with orientation toward the origin
Mathematics
1 answer:
Tomtit [17]3 years ago
4 0

Apparently my answer was unclear the first time?

The flux of <em>F</em> across <em>S</em> is given by the surface integral,

\displaystyle\iint_S\mathbf F\cdot\mathrm d\mathbf S

Parameterize <em>S</em> by the vector-valued function <em>r</em>(<em>u</em>, <em>v</em>) defined by

\mathbf r(u,v)=7\cos u\sin v\,\mathbf i+7\sin u\sin v\,\mathbf j+7\cos v\,\mathbf k

with 0 ≤ <em>u</em> ≤ π/2 and 0 ≤ <em>v</em> ≤ π/2. Then the surface element is

d<em>S</em> = <em>n</em> • d<em>S</em>

where <em>n</em> is the normal vector to the surface. Take it to be

\mathbf n=\dfrac{\frac{\partial\mathbf r}{\partial v}\times\frac{\partial\mathbf r}{\partial u}}{\left\|\frac{\partial\mathbf r}{\partial v}\times\frac{\partial\mathbf r}{\partial u}\right\|}

The surface element reduces to

\mathrm d\mathbf S=\mathbf n\,\mathrm dS=\mathbf n\left\|\dfrac{\partial\mathbf r}{\partial u}\times\dfrac{\partial\mathbf r}{\partial v}\right\|\,\mathrm du\,\mathrm dv

\implies\mathbf n\,\mathrm dS=-49(\cos u\sin^2v\,\mathbf i+\sin u\sin^2v\,\mathbf j+\cos v\sin v\,\mathbf k)\,\mathrm du\,\mathrm dv

so that it points toward the origin at any point on <em>S</em>.

Then the integral with respect to <em>u</em> and <em>v</em> is

\displaystyle\iint_S\mathbf F\cdot\mathrm d\mathbf S=\int_0^{\pi/2}\int_0^{\pi/2}\mathbf F(x(u,v),y(u,v),z(u,v))\cdot\mathbf n\,\mathrm dS

=\displaystyle-49\int_0^{\pi/2}\int_0^{\pi/2}(7\cos u\sin v\,\mathbf i-7\cos v\,\mathbf j+7\sin u\sin v\,\mathbf )\cdot\mathbf n\,\mathrm dS

=-343\displaystyle\int_0^{\pi/2}\int_0^{\pi/2}\cos^2u\sin^3v\,\mathrm du\,\mathrm dv=\boxed{-\frac{343\pi}6}

You might be interested in
What equation is graphed in this figure?
Ksivusya [100]

to get the equation of any straight line, we simply need two points off of it, let's use those two points in the picture below.

(\stackrel{x_1}{0}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{5}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{5}-\stackrel{y1}{2}}}{\underset{run} {\underset{x_2}{1}-\underset{x_1}{0}}}\implies \cfrac{3}{1}\implies 3 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_2=m(x-x_2) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_2}{5}=\stackrel{m}{3}(x-\stackrel{x_2}{1})

keeping in mind that for the point-slope form, either point will do, in this case we used the second one, but the first one would have worked just the same.

7 0
1 year ago
Due to the increased gasoline surcharge for its deliveries, the Sunnyside Beach Snack Stand had to offset its losses by passing
schepotkina [342]

Answer:

5 and 88 cenct

Step-by-step explanation:

8 0
3 years ago
Hi all i really need help this is multiple choice
Amanda [17]
Supplementary angles
4 0
3 years ago
Read 2 more answers
David had a coupon for the grocery store. He started graphing on the x-axis what his total bill would be if he didn’t use the co
Gre4nikov [31]

Answer:

Answer is 68$

Step-by-step explanation:

4 0
3 years ago
Estimate: 67times85 plzz help me out with my home work!!
natta225 [31]
5,695= 67x85 that's the answer
7 0
3 years ago
Other questions:
  • If X = 7 units, Y = 4 units, Z = 18 units, and h = 7 units, what is the surface area of the triangular prism shown above?
    9·1 answer
  • Algebra 2 Common Core
    11·1 answer
  • Cheryl workers 43.75 hours this week at a rate of $8.15 per hour. She gets paid overtime(time and a half) for any hours over 40.
    11·1 answer
  • Can someone help me answer this!
    13·2 answers
  • All you need is in the photo ​
    8·2 answers
  • Which statement about the graph of y = }}) * is true?
    14·1 answer
  • Cuánto es -2+(-3)+4-(-3)-5
    13·2 answers
  • PLZ HELP ASAPPPP PLEASE
    9·1 answer
  • Based on the scatterplot below, what is the most likely value for “Pounds of Apples” when “Number of Apples” is 80? A graph titl
    9·2 answers
  • Hello, can you help me? I don't speak English, I speak Spanish and I don't understand anything. I use a translator to communicat
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!