An inconsistent conservative vector field

In summary, the given vector field is conservative and path independent due to the parametric equations having continuous first order derivatives and the curl being <0, 0, 0>. However, in computing the scalar field equation, a computation error resulted in two different answers. The correct function is [f(x,y,z)] = xy^2 + ye^(3z), and the error was found in the last integral in the z-direction.
  • #1
riemannsigma
10
0
< 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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  • #2
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?
 
  • #3
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
 
  • #4
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??
 
  • #5
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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Likes riemannsigma
  • #6
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
 

FAQ: An inconsistent conservative vector field

What is an inconsistent conservative vector field?

An inconsistent conservative vector field is a type of vector field in which the curl of the vector field is not equal to zero, meaning that the vector field is not conservative. In other words, the vector field does not satisfy the fundamental theorem of calculus, which states that the line integral of a conservative vector field depends only on the endpoints and not the path taken.

What does it mean for a vector field to be conservative?

A conservative vector field is one in which the line integral (or work done) around a closed path is equal to zero. In other words, the path taken does not affect the total work done by the vector field.

Why is an inconsistent conservative vector field important?

An inconsistent conservative vector field is important because it can lead to incorrect calculations and predictions in physics and engineering. It also highlights the limitations and assumptions of vector calculus and the fundamental theorem of calculus.

How can you identify an inconsistent conservative vector field?

To identify an inconsistent conservative vector field, you can calculate the curl of the vector field and check if it is equal to zero. If the curl is not equal to zero, then the vector field is not conservative. You can also plot the vector field and observe if the field lines form closed loops.

Can an inconsistent conservative vector field be made into a conservative one?

No, an inconsistent conservative vector field cannot be made into a conservative one. This is because the inconsistency in the curl cannot be fixed or corrected. However, the vector field can be modified or adjusted to better fit the data or observations, but it will still not be a true conservative vector field.

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