{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

010.104-2005-2-fin

# 010.104-2005-2-fin - t ≤ 1 6 Let F =-GmM r/r 3 be the...

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

1. (15 points) Find the area of the surface cut from the hemisphere x 2 + y 2 + z 2 = 4 , z 0, by the cylinder x 2 + y 2 = 2 x . 2. (15 points) Let V = x 2 y + 1 3 y 3 i + x j . Among all smooth simple closed curves in the plane, oriented counter- clockwise, find the maximum circulation of V (i.e., R C V · d s ). 3. (15 points) Find the area inside the loop of the folium of Descartes described by the equation x 3 + y 3 = 3 axy for a > 0. ( Hint . The folium has a parametric representation x = 3 at 1 + t 3 , y = 3 at 2 1 + t 3 , where the loop is traversed counterclockwise for 0 t < .) 4. (15 points) Evaluate the integral RR S ( ∇× F ) · d S , where S is the portion of the surface of a sphere defined by x 2 + y 2 + z 2 = 1 and x +2 y +2 z 1, and where F = r × ( i + j + k ), r = x i + y j + z k . (Recall that we agreed to orient the sphere with the outward unit normal.) 5. (15 points) Let F = ( e x sin y ) i +( e x cos y ) j + z 2 k . Evaluate the integral R c F · d s , where c ( t ) = ( t, πt 3 / 2 , exp t ), 0

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: t ≤ 1. 6. Let F =-GmM r /r 3 be the gravitational force ﬁeld deﬁned on R 3 \{ } , where r = x i + y j + z k and r = k r k . (a) (5 points) Show that ∇ · F = 0. (b) (10 points) Show that F 6 = ∇ × G for any C 1 vector ﬁeld G on R 3 \ { } . 1 7. (15 points) Calculate the outward ﬂux of F = x 2 i-2 xy j +3 xz k across the boundary of the region W which is cut from the ﬁrst octant ( x ≥ ,y ≥ ,z ≥ 0) by the sphere x 2 + y 2 + z 2 = 4. 8. (15 points) Let C be the ellipse ( x-1) 2 + 4 y 2 = 4, and let f ( x,y ) = ln p x 2 + y 2 . Calculate Z C ∂f ∂n ds, where ∂f/∂n is the directional derivative of f in the direction of n (the outward unit normal on C ). 2...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

010.104-2005-2-fin - t ≤ 1 6 Let F =-GmM r/r 3 be the...

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

View Full Document
Ask a homework question - tutors are online