An inconsistent conservative vector field

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Discussion Overview

The discussion revolves around the properties of a specific vector field, , and its classification as conservative. Participants explore the implications of this classification, particularly regarding path independence in line integrals, and the discrepancies encountered when evaluating integrals along different paths.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the vector field is conservative because it is defined in a simply connected domain and has a curl of <0, 0, 0>.
  • Participants report obtaining different scalar field equations when integrating along various paths, leading to confusion about path independence.
  • One participant presents a specific computation along the x, y, and z axes, questioning where the error occurred in their calculations.
  • Another participant introduces a different vector field, , claiming it also exhibits path-dependent behavior, raising further questions about the nature of conservative fields.
  • Corrections are offered regarding missed terms in integrals, specifically a < -y_1 > term in one participant's z-direction integral.

Areas of Agreement / Disagreement

Participants generally agree that the vector field is conservative, but there is significant disagreement and confusion regarding the path independence of the line integrals, with multiple views on the source of discrepancies in results.

Contextual Notes

Participants express uncertainty about the computations leading to different scalar field equations and the implications of these results for the classification of the vector field. There are unresolved mathematical steps and potential computational errors that contribute to the confusion.

riemannsigma
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< y^2, 2xy+ e^(3z), 3ye^(3z)> is the vector field.

the above vector field is inside an open simply connected domain.

the parametric equations all have a continuous first order derivative inside the domain.

Lastly, the curl of the vector field is <0, 0, 0>

Thus, the vector field is conservative and is path independent.

HOWEVER...

If i take the

x axis, y axis, and z axis,
Y axis, x axis, and z axis, OR
Y axis, z axis, and x-axis

pathways, then I get the scalar field equation to be

[f(x,y,z)] = xy^2 + y + ye^(3z)

This is NOT CORRECT!

If I take the three axis pathways in any other order than the order listed above, then I get the correct answer.

Correct function is
[f(x,y,z)] = xy^2 + ye^(3z)There is not doubt that the vector field is conservative.

If a vector field is conservative, then the path integral MUST BE path independent.

Why do I get two different answers?...
 
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riemannsigma said:
< y^2, 2xy+ e^(3z), 3ye^(3z)> is the vector field.

the above vector field is inside an open simply connected domain.

the parametric equations all have a continuous first order derivative inside the domain.

Lastly, the curl of the vector field is <0, 0, 0>

Thus, the vector field is conservative and is path independent.

HOWEVER...

If i take the

x axis, y axis, and z axis,
Y axis, x axis, and z axis, OR
Y axis, z axis, and x axis

pathways, then I get the scalar field equation to be

[f(x,y,z)] = xy^2 + y + ye^(3z)

This is NOT CORRECT!

If I take the three axis pathways in any other order than the order listed above, then I get the correct answer.

Correct function is
[f(x,y,z)] = xy^2 + ye^(3z)There is not doubt that the vector field is conservative.

If a vector field is conservative, then the path integral MUST BE path independent.

Why do I get two different answers?...
Probably because you made a computation error. How can we tell without seeing the computation?
Can you show how you got ##xy²+y+ye^{3z}## in one case and ##xy²+ye^{3z}## in an other case?
 
X axis- y-axis - z axis pathway. 1. x-axis
y,z,dy,dz=0

Interval is from x=0 to x=x1

Integrate <0, 1, 0> • <dx, 0, 0> to get 0

2. Y-axis

x=x1, dx,dz,z=0

Interval is from y=0 to y=y1

Integrate <y^2, 2x1y + 1, 3y> • <0, dy, 0> to get
x1y^2 + y

3. Z-axis
x=x1, y=y1, dx=dy=0

Interval is from z = 0 to z = z1
Integrate <(y1)^2, 2x1y1 + e^(3z1), 3y1e^(3z)> • <0,0,dz> to get

y1e^(3z1)The total path integral is NOT CORRECT!

[f(x1, y1, z1)] = Constant + x1y^2 + y + y1e^(3z1)Where did I go wrong?? Help
 
Also... The same inconsistency occurs for this conservative vector field.

<e^y , xe^y>

Having e in the parametric equation seems to make the line integral path DEPENDENT.

WHATS GOING ON??
 
riemannsigma said:
X axis- y-axis - z axis pathway.1. x axis
y,z,dy,dz=0

Interval is from x=0 to x=x1

Integrate <0, 1, 0> • <dx, 0, 0> to get 0

2. Y-axis

x=x1, dx,dz,z=0

Interval is from y=0 to y=y1

Integrate <y^2, 2x1y + 1, 3y> • <0, dy, 0> to get
x1y^2 + y

3. Z-axis
x=x1, y=y1, dx=dy=0

Interval is from z = 0 to z = z1
Integrate <(y1)^2, 2x1y1 + e^(3z1), 3y1e^(3z)> • <0,0,dz> to get

y1e^(3z1)The total path integral is NOT CORRECT!

[f(x1, y1, z1)] = Constant + x1y^2 + y + y1e^(3z1)Where did I go wrong?? Help
In the last integral in the z-direction you miss a ##-y_1## term.
##\displaystyle \int_0^{z_1} 3y_1e^{3z} \, dz = \left. y_1e^{3z} \right|_0^{z_1} = y_1e^{3z_1}-y_1##
 
Last edited:
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Samy_A said:
In the last integral in the z-direction you miss a ##-y_1## term.
##\displaystyle \int_0^{z_1} 3y_1e^{3z} \, dz = \left. y_1e^{3z} \right|_0^{z_1} = y_1e^{3z_1}-y_1##
Yes! Lol. Thanks
 

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