calc_tutorial_16_5_023

 
Problem Statement

{2,4,5,7,8,10}
2*8

Let v = 8zk be the velocity field (in meters per second) of a fluid in R3. Calculate the flow rate (in cubic meters per second) through the upper hemisphere (z ≥ 0) of the sphere x2 + y2 + z2 = 1.

 
Step 1

Question Sequence

Question 1

Recall how to find flow rate through a surface, S, for a fluid with velocity vector field v. Choose the correct integral below to calculate the flow rate across the S (volume per unit time).

A.
B.
C.

To calculate the integral, you will also need to recall how to find a vector surface integral. Let F be a vector field. Let G(u, v) be an oriented parametrization of an oriented surface S with parameter domain D. Assume that G is one-to-one, and regular, except possibly at points on the boundary of D. Then we can calculate the vector surface integral as follows.

A.
B.
C.

If the orientation of S is reversed, the surface integral changes sign.

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 2

Before we can make any calculations, we must find an appropriate parametrization for the upper hemisphere of the sphere x2 + y2 + z2 = 1 since that is the surface through which the fluid is moving. Since this is part of a sphere, it will be easiest to convert the surface to spherical coordinates.

State the standard conversions.

x = R cos θ

y = R sin θ

z = R

Because the surface through which the fluid is moving is only the upper hemisphere, we have the following bounds on the parameters θ and φ.

0 ≤ θ <

A.
B.
C.
D.

0 ≤φ ≤

A.
B.
C.
D.
Correct.
Incorrect.

 
Step 3

Question Sequence

Question 3

Since the radius of the sphere is 1, we have the following parametrization.

G(θ, φ) = (cos θ sin φ, sin θ sin φ, cos φ)

Find Tφ and Tθ for G(θ, φ). Then use these vectors to compute n(φ, θ), the outward normal vector through the upper hemisphere.

Tφ = (∂G / ∂φ) =

A.
B.
C.

Tθ = (∂G / ∂θ) =

A.
B.
C.

n(φ, θ) = Tφ × Tθ =

A.
B.
C.
Correct.
Incorrect.

 
Step 4

Question Sequence

Question 4

Express the velocity field v = 8zk in terms of the parameters φ and θ. In other words, calculate v(G(θ, φ)).

v = 8zk

= <0, 0, 8z>

Therefore v(G(θ, φ)) = <, , cosφ>

Correct.
Incorrect.

 
Step 5

Question Sequence

Question 5

Compute the dot product v(G(θ, φ)) ·n(φ, θ).

v(G(θ, φ)) · n(φ, θ) = <0, 0, 8 cos φ> · sin φ <cos θ sin φ, sin θ sin φ, cos φ>

= sin φ cos2φ

Correct.
Incorrect.

 
Step 6

Question Sequence

Question 6

Integrate to calculate the flow rate through the upper hemisphere.

A.
B.
C.
D.
Correct.
Incorrect.