To check whether a vector field depends on the path just check if the rotation is equal to 0 ?

How can I check if the integral curvilinear depends on the path ?

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- Jun 8th 2009, 09:05 AMApprentice123Dependence of the pathTo check whether a vector field depends on the path just check if the rotation is equal to 0 ?

How can I check if the integral curvilinear depends on the path ?

- Jun 8th 2009, 09:17 AMRuun
If $\displaystyle \nabla \times F = 0$ then F is a conservative field, and integral doesn't depend on the path, only on the initial and final position. Also the reciprocal is true

- Jun 8th 2009, 09:19 AMAmer
f is independent of path in Q if

1)$\displaystyle \oint F.dr =0 $ for all simple closed curve in Q piece wise smooth

2)F is conservative (in Q) is gradient

3)curl F = o for all (x,y,z) in Q

if one is true then all is true

all the statement are equivalent - Jun 8th 2009, 09:34 AMApprentice123
- Jun 8th 2009, 10:15 AMApprentice123
Correct the example ?

- Jun 8th 2009, 10:18 AMAmer
that what my professor said to us independent of path mean the curl of the vector equal zero

but I didn't see it in the book of calculus 3) the curl is zero

this from my professor so I do not know if it is true or not but 1) and 2)

but I think it is true take this example

show that this is independent of path

$\displaystyle F(x,y)=2xy^3i + (1+2x^2y^2)j $ is independent of path

the curl is zero or as the book solve it

$\displaystyle since f(x,y)=2xy^3$ and $\displaystyle g(x,y)=(1+3x^2y^2)$

$\displaystyle \frac{\partial g}{\partial x } = 6xy^2 = \frac{\partial f}{\partial y}$ so 6 holds for all (x,y)

see this

Conservative vector field - Wikipedia, the free encyclopedia

they said every conservative vector is irrotational irrotational mean the curl is zero - Jun 8th 2009, 10:45 AMApprentice123
- Jun 8th 2009, 12:01 PMRuun
If there exist a scalar function $\displaystyle f$ such that $\displaystyle F=\nabla f$ then $\displaystyle \nabla \times F = \nabla \times \nabla f = 0$

In Vector Calculos by Marsden & Tromba in the section 3.4 the Theorem: For every function $\displaystyle f \in C^2$ $\displaystyle \nabla \times \nabla f = 0 $

The demonstration is just computing $\displaystyle \nabla \times \nabla f $ and use that $\displaystyle \frac{\partial f^2}{dxdy} = \frac{\partial f^2}{dydx}$ and so on. - Apr 14th 2013, 01:39 PMmathlover10Re: Dependence of the path
so for example for a conservative force, divide the closed path from Po to P1, the work done on path c1 is equal to path c2 and since they are in opposite directions they sum to 0. is there any relationship to the physics sum of all paths integral? why are the conditions: positive oriented, piecewise smooth, simple closed curve necessary conditions for this and Green's Theorem?