hw5_solutions

Hw5_solutions - February 6th 2009 Math 20e Assignment#5 Solutions Problem 1(page 310#2 Find the divergence of the vector eld V(x y z = zy i xz j xy

This preview shows pages 1–3. Sign up to view the full content.

February 6th, 2009 Math 20e - Assignment #5 Solutions Problem 1 (page 310 #2) . Find the divergence of the vector ﬁeld V ( x,y,z ) = zy i + xz j + xy k . Solution. Let V ( x,y,z ) = h yz,xz,xy i = h V 1 ( x,y,z ) ,V 2 ( x,y,z ) ,V 3 ( x,y,z ) i . Then we com- pute the divergence by taking div V = ∇ · V = ∂V 1 ∂x + ∂V 2 ∂y + ∂V 3 ∂z = ∂x ( yz ) + ∂y ( xz ) + ∂z ( xy ) = 0 + 0 + 0 = 0 . Problem 2 (page 311 #7) . Sketch a few ﬂow lines for F ( x,y ) = y i . Calculate ∇ · F and explain why your answer is consistent with your sketch. Solution. ∇ · F = ∂F 1 ∂x + ∂F 2 ∂y = ∂x ( y ) + ∂y (0) = 0 . 1

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

View Full Document
F = ∇ · F = 0 is consistent with the graph because zero divergence means the vector ﬁeld should have no expansions or contractions, and the ﬂow lines do not spread out or get closer together. Problem 3 (page 311 #8) . Sketch a few ﬂow lines for F ( x,y ) = - 3 x i - y j . Calculate ∇· F and explain why your answer is consistent with your sketch. Solution. ∇ · F = ∂F 1 ∂x + ∂F 2 ∂y = ∂x ( - 3 x ) + ∂y ( - y ) = - 4 . div F = ∇ · F = - 4 is consistent with the graph because negative divergence means the vector ﬁeld should contract, and the ﬂow lines do not spread out or get closer together. Problem 4 (page 312 #26) . Show that F = ( x 2 + y 2 ) i - 2 xy j is not a gradient ﬁeld. Solution. If F was a gradient ﬁeld, then we could write F = f for some scalar function f . But then by Theorem 1 on page 303 we know that the curl of a gradient is always zero, which means 0 = ∇ × ∇ f = ∇ × F. 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/30/2010 for the course MATH 20E 20E taught by Professor Enright during the Fall '09 term at UCSD.

Page1 / 8

Hw5_solutions - February 6th 2009 Math 20e Assignment#5 Solutions Problem 1(page 310#2 Find the divergence of the vector eld V(x y z = zy i xz j xy

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online