New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here


by: Evert Christiansen


Evert Christiansen
Texas A&M
GPA 3.92


Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

Class Notes
25 ?




Popular in Course

Popular in Mathematics (M)

This 4 page Class Notes was uploaded by Evert Christiansen on Wednesday October 21, 2015. The Class Notes belongs to MATH 251 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/226053/math-251-texas-a-m-university in Mathematics (M) at Texas A&M University.

Similar to MATH 251 at Texas A&M

Popular in Mathematics (M)




Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/21/15
1 E0 00 U rh 03 MATH 251504 Practice Problems for the Final Examination Fall 2006 Use the Divergence Theorem to evaluate ffs F dS where Fy z 2zi 26y x23j 722 6y cos c k and S is the surface of the solid E that lies in the region z 2 0 and is bounded by the parabolic cylinder y 1 7 2 and the planes y 0 z 0 and z x Find curlF and div F a Fxyz ewyzi zgj cosy k b yizi2zjyizk c Fxyz 4i x 722j 4k WWW Determine whether or not F is conservative and if it is conservative nd a function f such that F Vf a Fxy 2yem2 i e2 cos yj b Fxyz ysinzixsinzj 5sycoszk C Mtg2 yzi j9cyk 2y y 1 Evaluate f0 1222 d5 where C is the line segment from 030 to 7121 Evaluate fCF dr where Fxyz lnxi yzj 3k and C is given by I39t 3itjt2kfor 72 gt 2 Use Green7s Theorem to evaluate f03x2y 12y2 dx 3 siny dy where C is the boundary of the trapezoid with vertices 00 20 02 and 24 with clockwise orientation 7 00 H E0 Use the Fundamental Theorem for Line Integrals to evaluate f0 F dr where Fxyz y4zi 4xy32 eyzj 6y 114 k and C is given by I39t costi sintj 2tj for 0 S t S 77139 Use Stokes7 Theorem to evaluate ffs curlF dS where x2y2 Z Wag72 iyz722jezk and S is the part of the sphere 2 y2 22 10 that lies below the plane 2 71 with upward orientation Solutions Using the Divergence Theorern7 1 17m2 x FdS dideV 26ydzdyd S E 0 0 0 1 142 24 1 142 1 y1m2 226g dyd 2zeydyd 2mg dx 0 0 10 0 0 0 110 1 2 1 617m 2x51 w2 7 2x dx 7 7 2 e 7 2 0 0 a curlF 7x siny 7 322 i zyemyz y sinyj 7 26M k divF yzemyz b curlF 7j 7 2k divF 71 C curlF 2 7 z i 2 7 z k divF 0 3 a Since 22y6w2 26m2 6m2 cosy and F is de ned on all of R2 F is conservative ie F Vf for some function f Since fwzy 22y6m2 we must have f2y ye2 gy for some single variable function 9 Then fyy 2 g y so that g y cos y in which case we may take 9y siny so that ay ye2 siny note that f is unique only up to addition of a constant term b Since curlF 0 and F is de ned on all of R3 F is conservative ie F Vf for some function f Since fmyz z ysinz we have fyz 22 xy sinz gyz for some two variable function 9 Then fyzyz xsinz gyz and so gyz 0 which means that gyz hz for some single variable function h Then fzyz xy cosz h z and thus h z 5 so that we may take My 52 and hence fyz zz xy sinz 52 note that f is unique only up to addition of a constant term c Since curlF lt2 0 72gt which is not zero everywhere F is not conservative r Pararnetrize C by I39t lt7t3 7 ttgt for 0 S t S 1 Then r t lt71711gt so that lr tl xZ and hence 1 3 1 1 17 3 was lt3 7 t2t2 dt xglt3t3 7 a4 25 i C 0 2 5 0 10 5 We have r t lt012tgt and hence 2 2 1 2 Fdr ltln3t33gt lt012tgtdt 9 6tdt it 3t2 0 c 72 72 4 2 6 Using Green7s Theorem 2 22 2 21 322 122 2 dx zssin d 7242 d dx 712 2 dx y y y y y y y C 0 0 0 110 2 712953 i 48952 i 4895 dz 0 2 73954 71623 7 24952 7272 0 5 Since F Vf where fyz xy4z 6 as can be found by a procedure similar to that in the solution of 3b7 we have7 by the Fundamental Theorem for Line Integrals7 CF dr fr77r 7 fr0 f710147r7 f100 147r The boundary 0 of S is a Circle which we pararnetrize by I39t lt3 cos t 3sin t 71gt for 0 S t S 27139 Then r t lt73 sin t73cos t0gt Thus7 by Stokes7 Theorern7 27r CurlFdSFdr lt797971egtlt73sint73cost70gtdt S C 0 27r 27f 27sintcos t dt 277cost sin 0 0 0 0


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Steve Martinelli UC Los Angeles

"There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Janice Dongeun University of Washington

"I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.