# Parametrizing the portion

## Homework Statement

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

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?

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SammyS
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## Homework Statement

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

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 ?

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

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

0 < r < 2, 0 < θ < 2∏

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SammyS
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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 .

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

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.

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SammyS
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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 .

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
Staff Emeritus