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
kari74 [83]
3 years ago
6

Compute the differential of surface area for the surface S described by the given parametrization.

Mathematics
1 answer:
AysviL [449]3 years ago
8 0

With S parameterized by

\vec r(u,v)=\langle e^u\cos v,e^u\sin v,uv\rangle

the surface element \mathrm dS is

\mathrm dS=\left\|\dfrac{\partial\vec r}{\partial u}\times\dfrac{\partial\vec r}{\partial v}\right\|\,\mathrm du\,\mathrm dv

We have

\dfrac{\partial\vec r}{\partial u}=\langle e^u\cos v,e^u\sin v,v\rangle

\dfrac{\partial\vec r}{\partial v}=\langle -e^u\sin v,e^u\cos v,u\rangle

with cross product

\dfrac{\partial\vec r}{\partial u}\times\dfrac{\partial\vec r}{\partial v}=\langle ue^u\sin v-ve^u\cos v,-ve^u\sin v-ue^u\cos v,e^{2u}\cos^2v+e^{2u}\sin^2v\rangle

\dfrac{\partial\vec r}{\partial u}\times\dfrac{\partial\vec r}{\partial v}=\langle e^u(u\sin v-v\cos v),-e^u(v\sin v+u\cos v),e^{2u}\rangle

with magnitude

\left\|\dfrac{\partial\vec r}{\partial u}\times\dfrac{\partial\vec r}{\partial v}\right\|=\sqrt{e^{2u}(u\sin v-v\cos v)^2+e^{2u}(v\sin v+u\cos v)^2+e^{4u}}

\left\|\dfrac{\partial\vec r}{\partial u}\times\dfrac{\partial\vec r}{\partial v}\right\|=e^u\sqrt{u^2+v^2+e^{2u}}

So we have

\mathrm dS=\boxed{e^u\sqrt{u^2+v^2+e^{2u}}\,\mathrm du\,\mathrm dv}

You might be interested in
Please answer please correct answer
andrey2020 [161]

Answer:

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
STEP 2 HELP ME ASAP !!!!
Whitepunk [10]

Answer:

y=7x

Step-by-step explanation:

x=time

8 0
3 years ago
(WILL MARK BRAINLIEST) Solve x^2-4x-12=0. SHOW YOUR WORK! show and DESCRIBE EACH STEP!
Natasha_Volkova [10]
<span>x^2 - 4x - 12 = 0
factors
(x - 6)(x + 2) = 0
</span>Set (x - 6) and (x + 2)<span> equal to </span><span>0</span><span> and solve for </span><span>xx</span>.<span>
x -  6 = 0; x = 6
x + 2 = 0; x = -2

answer
x = -2 or x = 6</span>
8 0
3 years ago
Read 2 more answers
Do this work If one do I will make brainliest her or him please
denpristay [2]

Answer:

If you need the explanation, just ask.

6 0
2 years ago
The dot plot shows the height, in feet, of a group of trees being studied. How many trees are being studied
TEA [102]
If you are reffering to this dot plot, it would be 13 trees being studied.

4 0
3 years ago
Other questions:
  • ASAP! PLEASE HELP MEEEE! I WILL GIVE BRAINLIEST!!!
    6·1 answer
  • 6 miles is approximately to 9 km.How many km Are equal to 54 miles.How many miles are equal to 12km
    11·1 answer
  • What is 75% as a fraction in simplest form
    14·1 answer
  • HELPP THIS ISNT MINE THIS MY SISAND I DONT LIKE LATH SOHELP PLEASE BECAUSE IT DUE NOW ASP!!!!
    5·2 answers
  • What are the values of a and b in the equation
    9·1 answer
  • Use the factor theorem to determine whether or not the first polynomial is a factor of the second
    10·1 answer
  • A rectangle has an area of 18 square units. What can be said of its height and width? Check all that apply.
    15·1 answer
  • Find the total price of a $20 item that has been marked up 40%.
    6·2 answers
  • Creation of reproductive cells (gametes) requires...​
    7·1 answer
  • Brainliest and show work on how to do all 3
    6·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!