Help<br><br> geometry<br><br> pls!!!!!!!!!!!

Let φ(x, y) = arctan (y/x) .

Alexandra [31]

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

8 0

2 years ago

The exponent indicates how many times the blank is used as a factor

Tamiku [17]

The exponent indicates how many times the number is used as a factor

7 0

3 years ago

Positive fractions less than 1 with an interval of 1/8 between each pair of fractions

lesya692 [45]

(1/8), (2/8), (3/8), (4/8), (5/8), (6/8), (7/8)

if u want, 2/8=1/4, 4/8=1/2, 6/8=3/4

3 0

3 years ago

Sunayana took a novel book from her school library. She

hram777 [196]

Answer:

4.1% ororor

Step-by-step explanation:

1/6 = 16.7%

3/8 = 37.5%

5/12 = 41.7%

= 95.9%

100% - 95.9% = 4.1%

8 0

2 years ago

In a city, taxicab fare is .80 cents for the first 1/4 mile and .20 for additional 1/4 mile. How far can you travel for $5.00?

kap26 [50]

Answer:

5 1/2 MILES

Step-by-step explanation:

$5.00-$.80=$4.20

4.20/.20=21/4= 5 1/4+1/4= 5 2/4 OR 5 1/2 MILES

Hope this help!

4 0

3 years ago

Read 2 more answers

Help geometry pls!!!!!!!!!!!