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
Varvara68 [4.7K]
3 years ago
10

The equation giving a family of ellipsoids is u = (x^2)/(a^2) + (y^2)/(b^2) + (z^2)/(c^2) . Find the unit vector normal to each

point of the surface of this ellipsoids.
Mathematics
1 answer:
Fynjy0 [20]3 years ago
3 0

Answer:

\hat{n}\ =\ \ \dfrac{\dfrac{x}{a^2}\hat{i}+\ \dfrac{y}{b^2}\hat{j}+\ \dfrac{z}{c^2}\hat{k}}{\sqrt{(\dfrac{x}{a^2})^2+(\dfrac{y}{b^2})^2+(\dfrac{z}{c^2})^2}}

Step-by-step explanation:

Given equation of ellipsoids,

u\ =\ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}

The vector normal to the given equation of ellipsoid will be given by

\vec{n}\ =\textrm{gradient of u}

            =\bigtriangledown u

           

=\ (\dfrac{\partial{}}{\partial{x}}\hat{i}+ \dfrac{\partial{}}{\partial{y}}\hat{j}+ \dfrac{\partial{}}{\partial{z}}\hat{k})(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2})

           

=\ \dfrac{\partial{(\dfrac{x^2}{a^2})}}{\partial{x}}\hat{i}+\dfrac{\partial{(\dfrac{y^2}{b^2})}}{\partial{y}}\hat{j}+\dfrac{\partial{(\dfrac{z^2}{c^2})}}{\partial{z}}\hat{k}

           

=\ \dfrac{2x}{a^2}\hat{i}+\ \dfrac{2y}{b^2}\hat{j}+\ \dfrac{2z}{c^2}\hat{k}

Hence, the unit normal vector can be given by,

\hat{n}\ =\ \dfrac{\vec{n}}{\left|\vec{n}\right|}

             =\ \dfrac{\dfrac{2x}{a^2}\hat{i}+\ \dfrac{2y}{b^2}\hat{j}+\ \dfrac{2z}{c^2}\hat{k}}{\sqrt{(\dfrac{2x}{a^2})^2+(\dfrac{2y}{b^2})^2+(\dfrac{2z}{c^2})^2}}

             

=\ \dfrac{\dfrac{x}{a^2}\hat{i}+\ \dfrac{y}{b^2}\hat{j}+\ \dfrac{z}{c^2}\hat{k}}{\sqrt{(\dfrac{x}{a^2})^2+(\dfrac{y}{b^2})^2+(\dfrac{z}{c^2})^2}}

Hence, the unit vector normal to each point of the given ellipsoid surface is

\hat{n}\ =\ \ \dfrac{\dfrac{x}{a^2}\hat{i}+\ \dfrac{y}{b^2}\hat{j}+\ \dfrac{z}{c^2}\hat{k}}{\sqrt{(\dfrac{x}{a^2})^2+(\dfrac{y}{b^2})^2+(\dfrac{z}{c^2})^2}}

You might be interested in
Which expression is equivalent to 24 + 20​
Maslowich
24+20=44
4(6+5)=44
the first one
8 0
3 years ago
What is 3/4f + 5 = -5
anyanavicka [17]
<span>3/4f + 5 = -5
Subtract 5 from both sides
3/4f= -10
Divide both sides by 3/4 so that the only thing remaining on the left side is the variable f.
Final Answer: f= 40/3 or 13 1/3 *Both answers are equivalent to each other.</span>
5 0
3 years ago
Read 2 more answers
(2+2)/2x4+16-10x2/2x3=x
vovikov84 [41]
(2+2):2x4:16-10x2:2x3=x
4:2x4+16-10x1x3=x
8+16-30=x
-6=x
x=-6

This signing : signifícate division


I help the my answer can help you
5 0
3 years ago
Factor the polynomial expression 4x3 - 4.
Sloan [31]
<h3>Answer:  4(x - 1)(x^2 + x + 1)</h3>

==================================================

Work Shown:

4x^3 - 4

4(x^3 - 1)

4(x - 1)(x^2 + x + 1)

In the last step, I used the difference of cubes factoring formula which is

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

3 0
2 years ago
The length of one side of a square pond is 15 feet. The pond is surrounded by a 3 foot wide walkway. What is the total area of t
rewona [7]

Answer:

Area = 324 ft²

Step-by-step explanation:

Given

Length = 15ft

Walkway = 3ft

Required

Determine the area of the pond + the walkway

To calculate the required area, we need to get the total length of a side .

This is calculated by adding the length of a side of the pond and the side of the walkway

Total Length = Pond Length + Walkway

Total Length = 15ft + 3ft

Total Length = 18ft

Now, the required area can be calculated as thus:

Area = Total Length * Total Length

Area = 18ft * 18ft

Area = 324 ft²

8 0
2 years ago
Other questions:
  • Company B loses $1,575 for every employee who quits before 90 days. What is the total amount the company will lose if 2 employee
    12·1 answer
  • The sum of -15/32 and 7/32 is <br><br><br><br><br><br> please help ASAP
    15·2 answers
  • The sales tax on the table size is $9.23.
    7·1 answer
  • Simplify. y 5 • y 3 ÷ y 2
    10·2 answers
  • Math question please show work due today last day to submit Assignments or I fail please help
    14·2 answers
  • What is 10X995<br> please help
    6·2 answers
  • What is 2 over 3 +3 over 6​
    10·2 answers
  • Find the area of the shaded region. Round your answer to the nearest hundredth.
    7·2 answers
  • 19 points if uu help meeee
    13·2 answers
  • Use the interval notation to express the solution -6x&gt;42​
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!