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GreenPrint
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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.
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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