Parametrizing a Portion of a Sphere Above z = √3 Using Cylindrical Coordinates

[Stan12]

Homework Statement

Parametrize the portion of the sphere given by x² + y² + z² = 7 that lies above z = √3. (use cylindrical coordinates)

Homework Equations

x² + y² = r²

The Attempt at a Solution

x = rcosθ
y = rsinθ
z = z
z = √3
x² + y² + (√3)² = 7
x² + y² = 4
r²cos²θ + r² sin²θ = 4
r² (cos² + sin²θ) = 4
r = 2
Is this the correct way of parametrization?

 

[SammyS]

Homework Statement

Parametrize the portion of the sphere given by x² + y² + z² = 7 that lies above z = √3. (use cylindrical coordinates)

Homework Equations

x² + y² = r²

The Attempt at a Solution

x = rcosθ
y = rsinθ
z = z
z = √3
x² + y² + (√3)² = 7
x² + y² = 4
r²cos²θ + r² sin²θ = 4
r² (cos² + sin²θ) = 4
r = 2
Is this the correct way of parametrization?

Where is your parametrization ?

What's your final answer?

 

[Stan12]

Hmm. I guess my attempt to parametrize was totally incorrect, my question now is how do I parametrize this problem?

 

[Stan12]

z = √(7 - r²cos²θ - r²sin²θ)
s(r,θ) = (2cosθ, 2sinθ, √(7 - r² (cos²θ - sin²θ)) )

0 < r < 2, 0 < θ < 2∏

 

[SammyS]

z = √(7 - r²cos²θ - r²sin²θ)
r(r,θ) = (rcosθ, rsinθ, √(7 - r² (cos²θ - sin²θ)) )

You're using the symbol, r, for two different things.

Are you saying that position vector, (x, y, z), is given by:

(x, y, z) = (rcosθ, rsinθ, √(7 - r² (cos²θ - sin²θ)) )​

?

You need to restrict the values of r and θ .

 

[Stan12]

I apologize for the bad notation, I found that 0 < r < 2 and 0 < θ < 2∏ as restriction

 

[Stan12]

Ok. let me start from scratch, I'm going to set x = rcosθ , y = rsinθ and z = √(7 - (rcosθ)² - (rsinθ)²), cylindrical coordinates.

I found the restriction to r by setting x² + y² = r² in
portion of sphere given. which gives us r² + z² = 7

Now, it says that portion lies above a plane z = √3.
I plugged in z, r² + (√3)² = 7
and found that r = 2

and in cylindrical coordinates the restriction on θ = 2∏

Next, parametric of the portion, S(x,y) = (x, y, √(7 - x² - y²))

Now, I'm stuck here. I'm unsure if this is correct approach in trying to parametrize.

 

[SammyS]

After thinking this over, I believe that your position vector should be of the form: (r, θ, z) .

Writing x² + y² + z² = 7 in cylindrical coordinates gives: r² + z² = 7.

Solve that for either r or z .

 

[Stan12]

z = √7 - r²

s(x,y) = < x, y, (7 - x² - y²)^1/2

s(r,θ) = < rcosθ, rsinθ, (7 - (rcosθ)² - (rsinθ)²)^1/2

 

[SammyS]

What is sin²(θ) + cos²(θ) ?