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

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

View Full Document Right Arrow Icon
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
Background image of page 1

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

View Full DocumentRight Arrow Icon
Background image of page 2
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

This note was uploaded on 09/09/2009 for the course MATHMATICS 010-102 taught by Professor Cho during the Spring '09 term at Seoul National.

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 Right Arrow Icon
Ask a homework question - tutors are online