Visualizing x(u,v): A Solution Requested

  • Thread starter niall14
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In summary, the image of the function x(u,v) = (cos u*cos v, cos u*sin v, sin u) where x: U -> R^3 and u,v are elements of R^2 such that -pi/2 < u < pi/2 and -pi < v < pi, is a unit sphere with radius 1 centered at (0, 0, 0). This can be seen by using spherical coordinates to show that x^2 + y^2 + z^2 = 1.
  • #1
niall14
5
0
can somebody give a solution to wat the image of x(u,v)=(cos u*cos v,cos U*sin v, sin u) is? where

x:U ->R^3
u,v is an element of R^2 such that -pi/2 < u < pi/2, -pi < v < pi

help appreciated greatly
thank you
 
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  • #2
Hi niall14, welcome to PF :smile:

niall14 said:
can somebody give a solution to wat the image of x(u,v)=(cos u*cos v,cos U*sin v, sin u) is? where

x:U ->R^3
u,v is an element of R^2 such that -pi/2 < u < pi/2, -pi < v < pi

help appreciated greatly
thank you

It is the unit sphere.
Think of coordinates on earth, defined by latitude (u) and longitude (v).

Cheers!
 
  • #3
thanks, but how did i show it is a sphere, i can work out it is unit speed?
 
  • #4
niall14 said:
thanks, but how did i show it is a sphere, i can work out it is unit speed?

If you pick any u, and vary v, you'll find you have a circle of the form (r cos v, r sin v, z).
Just like any circle of constant latitude on earth.

Equivalenty you can find the circles through the poles with constant longitude (v).

Btw, with unit sphere I did not mean unit speed.
I meant it's a sphere with radius 1.
 
  • #5
niall14 said:
can somebody give a solution to wat the image of x(u,v)=(cos u*cos v,cos U*sin v, sin u) is? where

x:U ->R^3
u,v is an element of R^2 such that -pi/2 < u < pi/2, -pi < v < pi

help appreciated greatly
thank you
If (x, y, z) is in the image then x= cos(u)cos(v), y= cos(u)sin(v), z= sin(v) for some u and v.
[tex]x^2+ y^2+ z^2= cos^2(u)cos^2(v)+ cos^2(u)sin^2(v)+ sin^2(v)[/tex]
[tex]= (cos^2(u)+ sin^2(u))cos^2(v)+ sin^2(v)= cos^2(v)+ sin^2(v)= 1[/tex]

Since [itex]x^2+ y^2+ z^2= 1[/itex] the image is the surface of the ball with center (0, 0, 0) and radius 1.

As I Like Serena said, this parameterization is just "spherical coordinates",
[tex]x= \rho cos(\theta) sin(\phi)[/tex]
[tex]y= \rho sin(\theta) sin(\phi)[/tex]
[tex]z= \rho cos(\phi)[/itex]
with "u" instead of "[itex]\theta[/itex]", "[itex]\pi- v[/itex]" instead of [itex]\phi[/itex] (to change the cosine to sine and vice-versa) and [itex]\rho[/itex] set equal to 1.
 
  • #6
HallsofIvy said:
Since [itex]x^2+ y^2+ z^2= 1[/itex] the image is the surface of the ball with center (0, 0, 0) and radius 1.

Good call! :smile:
 

1. What does "x(u,v)" represent in this solution?

"x(u,v)" represents a mathematical function that maps two independent variables, u and v, to a dependent variable, typically denoted as x. This type of function is also known as a bivariate function.

2. Why is it important to visualize x(u,v)?

Visualizing x(u,v) allows us to better understand the relationship between the independent variables and the dependent variable. It can also help in identifying patterns or trends in the data, making it easier to interpret and analyze.

3. What methods can be used to visualize x(u,v)?

There are various methods that can be used to visualize x(u,v), including plotting the function on a graph, creating a contour plot, or using a three-dimensional plot. Other methods such as heat maps, scatter plots, and surface plots can also be used depending on the specific data and purpose of the visualization.

4. Are there any limitations to visualizing x(u,v)?

Yes, there can be limitations to visualizing x(u,v) as it depends on the complexity of the function and the amount of data available. Some methods may not accurately represent the data or may not be suitable for certain types of functions. It is important to carefully choose the visualization method to ensure accurate and meaningful results.

5. How can visualizing x(u,v) be useful in scientific research?

Visualizing x(u,v) can be useful in scientific research as it allows researchers to gain insights into the data and identify relationships that may not be apparent through other methods. It can also help in communicating findings to others and in making predictions based on the data. Additionally, visualizing x(u,v) can aid in the development of new theories and hypotheses to further advance scientific understanding.

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