{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Homework7

# Homework7 - given by v = i x j z k in meters/second How...

This preview shows pages 1–2. Sign up to view the full content.

1 Mathematics 1c: Homework Set 7 Due: Monday, May 24 at 10am. 1. (10 Points) Section 7.3, Exercise 6 . Find an expression for a unit vector normal to the surface x = 3 cos θ sin φ, y = 2 sin θ sin φ, z = cos φ for θ in [0 , 2 π ] and φ in [0 , π ] . 2. (15 Points) Section 7.3, Exercise 15 (a) Find a parameterization for the hyperboloid x 2 + y 2 - z 2 = 25 . (b) Find an expression for a unit normal to this surface. (c) Find an equation for the plane tangent to the surface at ( x 0 , y 0 , 0) , where x 2 0 + y 2 0 = 25 . (d) Show that the pair of lines ( x 0 , y 0 , 0) + t ( - y 0 , x 0 , 5) and ( x 0 , y 0 , 0) + t ( y 0 , - x 0 , 5) lie in the surface and as well as in the tangent plane found in part (c) . 3. (10 Points) Section 7.4, Exercise 6 . Find the area of the portion of the unit sphere that is cut out by the cone z p x 2 + y 2 . 4. (10 Points) Section 7.5, Exercise 2 . Evaluate ZZ S xyz dS where S is the triangle with vertices (1 , 0 , 0) , (0 , 2 , 0) and (0 , 1 , 1) . 5. (10 Points) Section 7.6, Exercise 7 . Calculate the integral ZZ S F · d S , where S is the surface of the half-ball x 2 + y 2 + z 2 1 , z 0 , and where F = ( x + 3 y 5 ) i + ( y + 10 xz ) j + ( z - xy ) k . 6. (10 Points) Section 7.6, Exercise 15. Let the velocity field of a fluid be

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: given by v = i + x j + z k in meters/second. How many cubic meters of ﬂuid per second are crossing the surface x 2 + y 2 + z 2 = 1 ,z ≥ 0? (Distances are in meters.) 7. (15 Points) Section 7.6, Exercise 18 . If S is the upper hemisphere { ( x,y,z ) | x 2 + y 2 + z 2 = 1 ,z ≥ } oriented by the normal pointing out of the sphere, compute ZZ S F · d S for parts (a) and (b) . 2 (a) F ( x,y,z ) = x i + y j (b) F ( x,y,z ) = y i + x j (c) for each of the vector ﬁelds above, compute ZZ S ( ∇ × F ) · d S and Z C F · d S , where C is the unit circle in the xy plane traversed in the counterclock-wise direction (as viewed from the positive z axis). (Notice that C is the boundary of S . The phenomenon illustrated here will be studied more thoroughly in the next chapter, using Stokes’ theorem.)...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

Homework7 - given by v = i x j z k in meters/second How...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online