Solving Non-Conservative Vector Field Line Integrals

In summary, the fundamental theorem of line integrals states that a conservative vector field on an open connected region is defined by a potential function ψ such that F = \nablaψ. To test for a conservative vector field, one must first determine if there exists a potential function ψ such that F = \nablaψ. If so, then F is a conservative vector field.
  • #1
GreenPrint
1,196
0
Hi,

I'm studying calculus 3 and am currently learning about conservative vector fields.

=============================
Fundamental Theorem for Line Integrals
=============================
Let F be a a continuous vector field on an open connected region R in [itex]ℝ^{2}[/itex] (or D in [itex]ℝ^{3}[/itex]). There exists a potential function ψ where F = [itex]\nabla[/itex]ψ (which means that F is conservative) if and only if

[itex]\int_{C}[/itex] F[itex]\bullet[/itex]T ds = [itex]\int_{C}[/itex] F[itex]\bullet[/itex]dr = ψ(B) - ψ(A)

for all points A and B in R and all smooth oriented curves C from A to B.

==========================
Test for Conservative Vector Fields
==========================

Let F = <f,g,h> be a vector field defined on a connected and simply connected region of [itex]ℝ^{3}[/itex], where f,g, and h have continuous first partial derivatives on D. Then, F is a conservative vector filed on D (there is a potential function ψ such that F = [itex]\nabla[/itex]ψ) if and only if

[itex]f_{y}[/itex] = [itex]g_{x}[/itex], [itex]f_{z}[/itex] = [itex]h_{x}[/itex], and [itex]g_{z}[/itex] = [itex]h_{y}[/itex].

For vector fields in [itex]ℝ^{2}[/itex], we have the single condition [itex]f_{y}[/itex] = [itex]g_{x}[/itex].

==
Q. I'm also studying elementary differential equations and it looks to me like I can take any non conservative vector field and make it into a conservative vector field by finding a integration factor and then finding ψ. Once I find ψ I can just use the fundamental theorem of line integrals and evaluate it at two points. This method, if I can do this, appears to me to be a more easier way of evaluated non conservative vector field line integrals.

It appears to me that the following is true:

f(x,y,z) = [itex]ψ_{x}[/itex]
g(x,y,z) = [itex]ψ_{y}[/itex]
h(x,y,z) = [itex]ψ_{z}[/itex]

Given some arbitrary vector field, F(f(x,y,z),g(x,y,z),h(x,y,z)), and I find that
[itex]f(x,y,z)_{y}[/itex] ≠ [itex]g(x,y,z)_{x}[/itex]
or
[itex]ψ_{xy}[/itex] ≠ [itex]ψ_{yx}[/itex]

then shouldn't I be able to multiply by some function μ(x) to make the statement true

f(x,y,z)μ(x) and g(x,y,z)μ(x)
or
[itex]ψ_{x}[/itex]μ(x) and [itex]ψ_{y}[/itex]μ(x)

then
[itex]f(x,y,z)_{y}[/itex]μ(x) = [itex]g(x,y,z)μ(x)_{x}[/itex]
or
[itex]ψ_{xy}[/itex]μ(x) = [itex](ψ_{y}μ(x))_{x}[/itex]

Can't I then proceed to find μ(x) and then use the fact that
f(x,y,z)μ(x) = [itex]ψ_{x}[/itex]μ(x)
and solve for ψ

ψ = [itex]\int[/itex] [itex]ψ_{x}[/itex]μ(x) dx = [itex]f(x,y,z)_{2}[/itex] + f(y,z)
or
ψ = [itex]\int[/itex] [itex]f(x,y,z)[/itex]μ(x) dx = [itex]f(x,y,z)_{2}[/itex] + f(y,z)

and then proceed to solve for f(y,z) some how?

I feel as if there's some way to make any non conservative vector filed equation into a conservative vector valued function and then just apply the Fundamental Theorem for Line Integrals some how but I'm not exactly sure I'm solving for it correctly. Thanks for any help. I hope to learn how to solve such a differential equation.
 
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  • #2


I think there may be some non-conservative fields that could be represented by a conservative part and a non-conservative part but to say that non-conservative vector fields can be mapped to conservative ones is probably not true at all.

From the definition the conservative vector field can be represented by a potential whose gradient is the conservative vector field right? so given a potential function of x,y,z what could I add to it to make things no longer conservative say a time varying component. physically that makes it non-conservative right?

I'm thinking: gravity about the sun is conservative but if you make the sun's mass somehow vary over time then that would make it non-conservative. depending on the size and periodicity of the variations (if they are even periodic) so while you could represent the potential as some time varying function the grav force field wouldn't be conservative.

I think General Relativity has that issue and that why Mercury precesses in it correctly.

Im real rusty here someone please slap me.
 
  • #3


Well I don't see why it couldn't be done if you had a vector field that had only only two variables without a problem at all.

If I had a vector field
F = <f(x,y),g(x,y)>

I would run into something like this

ψ = [itex]\int[/itex] [itex]f(x,y)[/itex]μ(x) dx = [itex]f(x,y)_{2}[/itex] + f(y)

then I could say

[itex]ψ_{y}[/itex] = [itex]f(x,y)_{2y}[/itex] + f'(y) = g(x,y)μ(x)
because g(x,y)μ(x) = [itex]ψ_{y}[/itex]
and solve for f(y)
f(y) = [itex]\int[/itex] [g(x,y) - [itex]f(x,y)_{2y}[/itex]] dy

then
ψ = [itex]f(x,y)_{2}[/itex] + [itex]\int[/itex] [g(x,y)μ(x) - [itex]f(x,y)_{2y}[/itex]] dy
 
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  • #4


ok so I figured out were to go from here
ψ = [itex]\int[/itex] [itex]f(x,y,z)[/itex]μ(x) dx = [itex]f(x,y,z)_{2}[/itex] + f(y,z)
to solve for ψ
it was easier than I thought.
As far as I know this is a perfectly fine thing to do? It doesn't show up in my calculus book so I just want to make sure. I can make any non conservative vector field into a conservative vector field by multiply by a integration factor and then just evaluating it from the limits of the integral by the fundamental therom of line integrals? I just want to make sure I don't see why not.
 
  • #5


i was arguing from the physics pt where a conservative vector field was represented by the grad of a potential field and then trying to see how the pot field could be altered. Gen Rel creates this situation and that's the explanation why mercury precesses Gen Rel describes a non-conservative vector field about the sun due to the interaction of x,y,z and t the spacetime metric.

but I am glad you solved your problem.
 

What is a non-conservative vector field?

A non-conservative vector field is a type of vector field in which the line integral along a closed path is dependent on the path taken. In other words, the value of the line integral will vary depending on the specific path chosen, unlike in a conservative vector field where the value of the line integral is independent of the path.

How do you solve a non-conservative vector field line integral?

To solve a non-conservative vector field line integral, you must first determine whether the vector field is conservative or non-conservative. If it is non-conservative, you will need to use the fundamental theorem of line integrals, which involves calculating the line integral using a potential function. If the vector field is conservative, you can simply use the fundamental theorem of calculus to calculate the line integral.

What is a potential function and how is it used to solve non-conservative vector field line integrals?

A potential function is a scalar function that is used to represent a vector field. In the case of a non-conservative vector field, the potential function is used to calculate the line integral by taking the derivative of the potential function with respect to the path. This allows for a simpler and more straightforward method of calculating the line integral in a non-conservative vector field.

Can a vector field be both conservative and non-conservative?

No, a vector field can only be either conservative or non-conservative, but not both. A vector field is considered conservative if the line integral along any closed path is equal to zero, while a vector field is considered non-conservative if the line integral along a closed path is not equal to zero.

Why is it important to understand how to solve non-conservative vector field line integrals?

Understanding how to solve non-conservative vector field line integrals is important because it allows us to calculate the work done by a non-conservative force, which is a crucial concept in physics and engineering. It also helps us understand the behavior of physical systems and make predictions about their behavior in different scenarios.

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