# What is Conservative field: Definition and 25 Discussions

In vector calculus, a conservative vector field is a vector field that is the gradient of some function. Conservative vector fields have the property that the line integral is path independent; the choice of any path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the vector field being conservative. A conservative vector field is also irrotational; in three dimensions, this means that it has vanishing curl. An irrotational vector field is necessarily conservative provided that the domain is simply connected.
Conservative vector fields appear naturally in mechanics: They are vector fields representing forces of physical systems in which energy is conserved. For a conservative system, the work done in moving along a path in configuration space depends only on the endpoints of the path, so it is possible to define a potential energy that is independent of the actual path taken.

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1. ### Electric field around a circuit

We know that E is conservative so the integral of E around a closed loop is zero. I know this helps us (in some way, that's why i'm asking) to calculate the total voltage drop around the complete circuit (which is zero). What exactly is "E" in the integral? For example, internet says "electric...
2. ### I Spring balance analysis from a system point of view

Hi, I was thinking about a simple scenario in the framework of Newton (classic) mechanics. Take a spring scale/balance fixed at one end (wall) with a body attached to the other end's hook. From an analysis point of view we can consider as "system" the spring scale + the wall + the body. Then...
3. ### MHB Multivariable conservative field

Goodnight, How can I find a conservative field F, such that ∫F ds = 0 without C being a closed path Can i have some examples ?
4. ### Work of a non conservative field

Homework Statement Compute the work of the vector field $$H: \mathbb{R^2} \setminus{(0,0}) \to \mathbb{R}$$ $$H(x,y)=\bigg(y^2-\frac{y}{x^2+y^2},1+2xy+\frac{x}{x^2+y^2}\bigg)$$ in the path $$g(t) = (1-t^2, t^2+t-1) with t\in[-1,1]$$ Homework Equations 3. The Attempt at a Solution [/B] So...
5. ### Figuring Out if A Force Field is Conservative or Not

Homework Statement There is a collection of different force fields, for example: $$F_{x}=ln z$$ $$F_{y}=-ze^{-y}$$ $$F_{z}=e^{-y}+\frac{x}{z}$$ We are supposed to indicate whether they are conservative and find the potential energy function. Homework Equations See Above The Attempt at a...
6. ### Conservative vector field problem

Homework Statement Determine for which real values of the parameter ##\alpha## the vector field given by ##F(x,y) = (\frac{2xy}{y-\alpha}, 2 - \frac{4x^2}{(y-\alpha)^2})## is conservative. For those values of ##\alpha##, calculate the work done along the curve of polar equation: ##\rho =...
7. ### Is potential energy only acquired in a *Conservative Field*?

Is potential energy gained or lost only in a conservative field, or when work is done against or by conservative forces?
8. ### Non Uniform Electric Field Lines Problem

Ok we know that the electric field(uniform or non uniform) is a conservative field. Imagine three horizontal electric field lines in '+X' direction separated by unequal distances let's say line 1 and 2 is separated by distance 'a' and line 2 and 3 is separted by some distance 'b' such that...
9. ### Question about conditions for conservative field

Question about conditions for conservative field In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is simply connected and open. Usually in textbooks there is not much explanations on why these...
10. ### My Mind going to blow Why Conservative field Path independ? by TMSxPh

Searching for 'Conservative fields intuition' comes up with a thread pretty high up on the list with this question. Which was closed years ago. " Hi I'm now reading about Vector fields, everything is clear and intuitive for me as curl divergence ..ect , except one simple thing that I'm...
11. ### Is the Given Force Field Conservative?

Homework Statement Verify the following force field is conservative. F = 2xcos2yi - (x2+1)sin2yj Homework Equations ∇xF=0 The Attempt at a Solution I'm pretty sure this is just a mistake in the book, but according to my calculations, this isn't a conservative force. I used the...
12. ### Intuition for conservative field

A scalar field is conservative, i.e. the line integral does not depend on the path taken, if it has a gradient. Now, can someone give me intuition behind why the gradient would have something to do with this? :) For me the gradient is merely a way of writing up the partial derivates as...
13. ### Is Curl F Non-Zero in a Conservative Field?

It seems there are some problems on the test. I find that Curl F is not zero, that means the field is not exact. Am I right?
14. ### Is every closed path integral zero?

Hi I was wondering, one way that a conservative field can be found is if the line integral of any closed path is 0. However, what if I have a non-conservative field, I travel in a circle in a clockwise manner back to my starting point, then travel along the same path in a counter clockwise...
15. ### Examples where rot F = 0 don't imply conservative field

Hey, all. Anyway, I've been looking at books and sources online, and the only counterexample to the wrongly stated theorem \nabla \times \mathbf{F} = 0 \Leftrightarrow \text{conservative vector field} seems to be \mathbf{F} = \left(\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2} \right), or other...
16. ### My Mind going to blow Why Conservative field Path independ?

Hi I'm now reading about Vector fields, everything is clear and intuitive for me as curl divergence ..ect , except one simple thing that I'm straggling with for the last 4 days! I searched Internet and a lot of math & physics books but in vain : Why Conservative field is Path independent...
17. ### Conservative Field Clarification

Hi All, As I understand it, a conservative field means that the energy expended by an outside agent in going between any two points is independent of the path so that the closed line integral of Edotdl is zero. This is presented in the study of electrostatics. It seems to me that you...
18. ### Are magnetic fields conservative field?

The title says it all. I've heard my professor saying that they are not conservative. I'm very surprised by this. If it is true then I'll think about all the implications it generates. By the way today was the class where we were introduced magnetic fields for the first time. Thanks!
19. ### Calculate Line Integral: Conservative Field Question with Circle | Homework Help

Homework Statement I need to calculate: \oint_{\Gamma} \vec{F}\cdot d \vec{r} where: \vec{F} = \frac{-y \vec{i} + x \vec{j}}{x^2+y^2} where \Gamma is the positive direction circle: a. x2 + y2 = 1 b. (x-2)2 + y2 = 1 Homework Equations \int_{C} \nabla f \cdot d \vec{r} =...
20. ### Conservative field over a domain

Homework Statement prove that: \vec{F} = \frac{-y^2}{(x-y)^2}\vec{i} + \frac{x^2}{(x-y)^2}\vec{j} is a conservative field, and find the potential in the domain: D: (x+5)^2 + y^2 \leq 9 Homework Equations ? The Attempt at a Solution Well, \frac{\partial P}{\partial y} =...
21. ### Simplifiyng the proof of conservative field using rottor properties

\vec{F}=(\frac{x}{\sqrt{x^2+y^2+z^2}},\frac{y}{\sqrt{x^2+y^2+z^2}},\frac{z}{\sqrt{x^2+y^2+z^2}}) \vec{r}=(x,y,z) |r|=\sqrt{x^2+y^2+z^2} \vec{F}=(\frac{x}{|r|},\frac{y}{|r|},\frac{z}{|r|}) so its F=\frac{r}{|r|} i need to prove that F is a conservative field where (x,y,z)...
22. ### Finding a function in conservative field

Homework Statement Find the non-zero function h(x) for which: field F(x,y) = h(x) [xsiny + ycosy] i + h(x) [xcosy - ysiny] j is conservative. The Attempt at a Solution curlF=0 d/dx [h(x) [xcosy - ysiny] ] - d/dy [h(x) [xsiny + y cos y] ] = 0 xcosy = ysiny ? I...
23. ### Continuous 2nd Partials a Substantial Requirement for Conservative Field?

It can be shown that if F has continuous 2nd partials, then div curl F = 0. According to Stoke's Theorem, the work done around a closed path C is equal to the flux integral of curl F on a surface sigma that has C as its boundary in positive orientation. However, this integral is equal to the...
24. ### Understand Significance of Conservative Fields: Inspire Me!

hi PF. What is exactly a conservative field? I know the mathematical definitions such as the existence of a scalar potential, the curl of the field equals 0 (irrotational), path independence etc. But I still don't get a physical understanding of such a field. What's the significance of...
25. ### Is E a Conservative Field? Analyzing Two Fields for Conservation

Can anybody help me on this question,Show whether or not the following fields are conservative: E=yz^2i+(Xz^2+2)j+(2xyz-1)k E=-(x^2-y^2)i-2xyj+4k I don't think if am right I want to start by comparing i follow by j and z,example yz^2=-(x^2-y^2)