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
belka [17]
3 years ago
10

Let φ(x, y) = arctan (y/x) .

Mathematics
1 answer:
Alexandra [31]3 years ago
8 0

Answer:

a) \large F(x,y)=(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2})

b) \large \mathbb{R}^2-\{(0,0)\}

c) the points of the form (x, -x) for x≠0

Step-by-step explanation:

a)

If φ(x, y) = arctan (y/x), the vector field F = ∇φ would be

\large F(x,y)=(\frac{\partial \phi}{\partial x},\frac{\partial \phi}{\partial y})

On one hand we have,

\large \frac{\partial \phi}{\partial x}=\frac{\partial arctan(y/x)}{\partial x}=\frac{-y/x^2}{1+(y/x)^2}=-\frac{y/x^2}{1+y^2/x^2}=\\\\=-\frac{y/x^2}{(x^2+y^2)/x^2}=-\frac{y}{x^2+y^2}

On the other hand,

\large \frac{\partial \phi}{\partial y}=\frac{\partial arctan(y/x)}{\partial y}=\frac{1/x}{1+(y/x)^2}=\frac{1/x}{1+y^2/x^2}=\\\\=\frac{1/x}{(x^2+y^2)/x^2}=\frac{x}{x^2+y^2}

So

\large F(x,y)=(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2})

b)

The domain of definition of F is  

\large \mathbb{R}^2-\{(0,0)\}

i.e., all the plane X-Y except the (0,0)

c)

Here we want to find all the points such that

\large (-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2})=(k,k)

where k is a real number other than 0.

But this means

\large -\frac{y}{x^2+y^2}=\frac{x}{x^2+y^2}\Rightarrow y=-x

So, all the points in the line y = -x except (0,0) are parallel to the vector field F, that is, the points (x, -x) with x≠ 0

You might be interested in
What is the answer to this math problem<br> 4-3=<br> 8-9=
ivann1987 [24]

Answer:

4-3=1

8-9=-1

Step-by-step explanation:

I hope this helps

3 0
3 years ago
A piece of rope 27m long is cut into two pieces so that one piece is four-fifths as long as the other. Find the length of each p
nata0808 [166]

Answer:

x+.8x=27

1.8x=27

x=15

4/5(15)=12

So 15 and 12

Step-by-step explanation:

8 0
3 years ago
Solving these quadratics for x using complete the square 3x^2-4x-1=0. Show your work.
UkoKoshka [18]
Answer: x = (sqrt(7) + 2)/3 and
x = ( – sqrt(7) + 2)/3

Explanation:

3x^2 - 4x - 1 = 0

Divide both sides by 3:

3x^2/3 - 4/3x - 1/3 = 0/3
x^2 - 4/3x - 1/3 = 0
x^2 - 4/3x = 1/3
x^2 - 4/3x + (2/3)^2 = 1/3 + (2/3)^2
(x - 2/3)^2 = 1/3 + 4/9
(x - 2/3)^2 = 7/9

Sqrt both sides:

x - 2/3 = sqrt (7/9)
x - 2/3 = |sqrt(7)/3|

Set x -2/3 = sqrt(7)/3

=> x = (sqrt(7) + 2)/3

Set x - 2/3 = - sqrt(7)/3

=> x = ( - sqrt(7) + 2)/3
6 0
3 years ago
Is 27.5 a rational number
laiz [17]
Yes because it has an odd number on the end.
6 0
3 years ago
13.On many multiple-choice tests, 1 point is given for each correct
lana [24]
C is the answer I believe

Hoped it helped
7 0
3 years ago
Other questions:
  • Two is a zero of the equation x3−x2−14x+24=0. Which factored form is equivalent to the equation?
    15·1 answer
  • What is 3/5 of 12? Please answer!
    9·1 answer
  • After t seconds, a ball tossed in the air from the ground level reaches a height of h feet given by the equation h = 144t-16t^2
    13·2 answers
  • Help me solve this please 33+15w=3w-w+4w
    6·1 answer
  • If the perimetes of the right angled triangle is 36m what is the length of its hypotenuse?
    13·1 answer
  • State all integer values of x in the interval (-2,5) that satisfy the following
    5·1 answer
  • What is the area of the base of a paper cone, in square inches, with a height of 15 inches and a volume of 900 cubic inches?
    6·1 answer
  • Select the values that make the inequality -n ≤ 6 true.
    11·1 answer
  • A population of a particular yeast cell develops with a constant relative growth rate of 0.4425 per hour. The initial population
    7·1 answer
  • Study island math pls helppp<br> I will love you if you answer
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!