Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 14: Q. 57 (page 1134)

Prove that, for any conservative vector field F(x, y),

∬R ∂F2∂x−∂F1∂ydA=0

for any simply connected region R⊂R2 whose boundary is smooth or piecewise smooth.

Short Answer

Expert verified

hbdfhbf

Step by step solution

01

Step 1. Given Information

j dvjd

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!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

S is the portion of the saddle surface determined by z = x2 − y2 that lies above and/or below the annulus in the xy-plane determined by the circles with radii

32and2

and centered at the origin.

F(x,y,z)=−xzi−yzj+z2k, where S is the cone with equation z=x2+y2between z=2,4, with n pointing outwards.

Calculus of vector-valued functions: Calculate each of the following.

∫r(t)dt,wherer(t)=eti+t3j−4k

Let Rbe a simply connected region in the xy-plane. Show that the portion of the paraboloid with equation z=x2+y2 determined by R has the same area as the portion of the saddle with equation z=x2-y2determined by R.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The result of integrating a vector field over a surface is a vector.

(b) True or False: The result of integrating a function over a surface is a scalar.

(c) True or False: For a region R in thexy-plane,dS=dA.

(d) True or False: In computing ∫Sf(x,y,z)dS, the direction of the normal vector is irrelevant.

(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then ∫Sf(x,y,z)dSmeasures the flow through S in the positive z direction determined by f (x, y, z).

(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then ∫SF(x,y,z).ndSmeasures the flow through S in the direction of n determined by the field F(x, y, z).

(g) True or False: In computing ∫SF(x,y,z).ndS,the direction of the normal vector is irrelevant.

(h) True or False: In computing ∫SF(x,y,z).ndS,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the dSterm.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free