{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 16-3 - Math 200 Spring 2010 Handout#25 Section 16-3...

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

Math 200, Spring 2010 Handout #25 Section 16-3 Conservative Vector Fields Independence of Path and Conservative Vector Fields We say that the integral C d F s is independent of path if the integral is the same for every path having the same initial and terminal points, that is, 1 2 d d = c c F s F s for any two paths ( ) 1 t c and ( ) 2 t c from point P to point Q . A vector field F with this property is called conservative . Definition: A curve is closed if its two endpoints are the same. Theorem: For any closed path C in domain D , C d F s is path independent in D if and only if 0 C d = F s integralloop . Note: This theorem says that the work done by a conservative vector field moving an object along a closed curve is 0. The Fundamental Theorem for Gradient Vector Fields If ϕ = ∇ F on a domain D , then for every oriented curve C in D with initial point P and terminal point Q , ( ) ( ) C C d d Q P ϕ ϕ ϕ = = F s s . Note the similarity to the Fundamental Theorem of Calculus, which gives the integral of a derivative as the original function evaluated at the endpoints of an interval: ( ) ( ) ( ) b a f x dx f b f a = . Since the gradient is just another type of derivative, the above theorem should seem plausible. Note: This theorem says that if a vector field F is a gradient vector field, then we have a nice and easy approach for computing line integrals, and thus the work done by a conservative force field. Note: This also says that the line integral of a gradient vector field depends only on the endpoints of the curve C , not the path taken, that is, it is path independent. Note: If C is closed and F is a gradient vector fields, then 0 C d = F s integralloop 1. Let F be the gradient of ( ) 2 , , x y z xy z ϕ = + . Compute the line integral of F along: (a) The line segment from ( ) ( ) 1,1,1 to 1,2,2 .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern