Chapter 14: Q. 34 (page 1141)

$F (x, y, z) = <\ln (x y + 1) + 5^{x} 3^{y} 2^{z}, 4 x z^{2}>, and C is the$ $b o u n d a r y o f t h e s q u a r e i n t h e p l a n e z = 6 a n d w i t h v e r t i c e s a t (2, 0, 6), (- 2, 0, 6), (2, 4, 6), a n d (- 2, 4, 6) .$

Short Answer

Expert verified

$L e t S b e a n o r i e n t e d, s m o o t h o r p i e c e w i s e - s m o o t h s u r f a c e b o u n d e d b y a c u r v e C . S u p p o s e t h a t i s a n o r i e n t e d u n i t n o r m a l v e c t o r o f S a n d C h a s a p a r a m e t r i z a t i o n t h a t t r a v e r s e s C i n t h e o u n t e r c l o c k w i s e d i r e c t i o n w i t h r e s p e c t t o n .$

Step by step solution

stokes theorem

$S t o k e s' T h e o r e m s t a t e s t h a t, " L e t S b e a n o r i e n t e d, s m o o t h o r p i e c e w i s e - s m o o t h s u r f a c e b o u n d e d b y a c u r v e C . S u p p o s e t h a t $ \ m a t h b f {n} $ i s a n o r i e n t e d u n i t n o r m a l v e c t o r o f S a n d C h a s a p a r a m e t r i z a t i o n t h a t t r a v e r s e s C i n t h e c o u n t e r c l o c k w i s e d i r e c t i o n w i t h r e s p e c t t o n . I f a v e c t o r f i e l d F i s d e f i n e d o n S, t h e n,$

hypothesis

$\int_{C} F (x, y, z) \cdot d r = \iint_{S} curl F (x, y, z) \cdot n d S =$ $H e n c e, t h e h y p o t h e s i s o f S t o k e s' T h e o r e m i s a s f o l l o w s, (1) T h e v e c t o r f i e l d F i s d e f i n e d a n d c o n t i n u o u s l y d i f f e r e n t i a b l e o n a n o p e n s e t c o n t a i n i n g t h e s u r f a c e S b o u n d e d b y C . (2) T h e b o u n d a r y c u r v e C i s s i m p l e, s m o o t h a n d c l o s e d . (3) T h e s u r f a c e S i s o r i e n t e d a n d s m o o t h .$

consider

$C o n s i d e r t h e f o l l o w i n g s u r f a c e : T h e s u r f a c e S i s t h e p y r a m i d w i t h v e r t i c e s a t (0, 0, 6), (2, 0, 0), (- 2, 0, 0), (0, 3, 0) a n d (0, - 3, 0) . T h e o b j e c t i v e i s t o s h o w t h a t S t o k e s' T h e o r e m d o e s n o t a p p l y i n t h i s c a s e . H e r e, n o t i c e t h a t, t h i s s u r f a c e S i s n o t s m o o t h o r p i e c e - w i s e s m o o t h s u r f a c e . T h i s i m p l i e s t h a t, S t o k e s' T h e o r e m d o e s n o t a p p l y b y (3) .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$F (x, y, z) = <\ln (x y + 1) + 5^{x} 3^{y} 2^{z}, 4 x z^{2}>, and C is the$ $b o u n d a r y o f t h e s q u a r e i n t h e p l a n e z = 6 a n d w i t h v e r t i c e s a t (2, 0, 6), (- 2, 0, 6), (2, 4, 6), a n d (- 2, 4, 6) .$

Short Answer

Step by step solution

stokes theorem

hypothesis

consider

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Theoretical and Mathematical Physics

Pure Maths

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

F(x,y,z)=ln(xy+1)+5x3y2z,4xz2, andCis theboundaryofthesquareintheplanez=6andwithverticesat(2,0,6),(-2,0,6),(2,4,6),and(-2,4,6).

Short Answer

Step by step solution

stokes theorem

hypothesis

consider

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Theoretical and Mathematical Physics

Pure Maths

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

$F (x, y, z) = <\ln (x y + 1) + 5^{x} 3^{y} 2^{z}, 4 x z^{2}>, and C is the$ $b o u n d a r y o f t h e s q u a r e i n t h e p l a n e z = 6 a n d w i t h v e r t i c e s a t (2, 0, 6), (- 2, 0, 6), (2, 4, 6), a n d (- 2, 4, 6) .$