How to Use the Change of Variables Theorem to Prove that Reversing the Path of Integration Yields a Negative...
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I need to use the Change of Variables Theorem (basically the Chain Rule), which states that:
Given a change of coordinates $F: mathbb{R}^n to mathbb{R}^n$ of the form $overrightarrow {u} = F ( overrightarrow{x} )$, the integral of $h$ over some region $F(R)$ converts as:
$int_{F(R)} h( overrightarrow {u})d{overrightarrow {u}} = int_Rh(F( overrightarrow {x}) |DET[DF]|d{ overrightarrow {x}}$
I need to prove that reversing the orientation of some path $gamma$ has the effect of changing the sign of the path integral $int_{gamma}alpha$ for some 1-Form $alpha$.
I think that I need to make a change of coordinates $ overrightarrow {u} = -overrightarrow {u}$, but from here I'm stuck.
Is this the right way to go about proving this? Or is there a better way?
calculus integration multivariable-calculus vector-fields chain-rule
asked Nov 18 at 6:19
Jackson Joffe
335
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up vote
0
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I need to use the Change of Variables Theorem (basically the Chain Rule), which states that:
Given a change of coordinates $F: mathbb{R}^n to mathbb{R}^n$ of the form $overrightarrow {u} = F ( overrightarrow{x} )$, the integral of $h$ over some region $F(R)$ converts as:
$int_{F(R)} h( overrightarrow {u})d{overrightarrow {u}} = int_Rh(F( overrightarrow {x}) |DET[DF]|d{ overrightarrow {x}}$
I need to prove that reversing the orientation of some path $gamma$ has the effect of changing the sign of the path integral $int_{gamma}alpha$ for some 1-Form $alpha$.
I think that I need to make a change of coordinates $ overrightarrow {u} = -overrightarrow {u}$, but from here I'm stuck.
Is this the right way to go about proving this? Or is there a better way?
calculus integration multivariable-calculus vector-fields chain-rule
asked Nov 18 at 6:19
Jackson Joffe
335
2
Let $gamma(a),gamma(b)$ be the start and endpoint of $gamma$ then consider $gamma(a+b-t)$ over [a,b] for the reverse path
– Fakemistake
Nov 18 at 6:27
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I need to use the Change of Variables Theorem (basically the Chain Rule), which states that:
Given a change of coordinates $F: mathbb{R}^n to mathbb{R}^n$ of the form $overrightarrow {u} = F ( overrightarrow{x} )$, the integral of $h$ over some region $F(R)$ converts as:
$int_{F(R)} h( overrightarrow {u})d{overrightarrow {u}} = int_Rh(F( overrightarrow {x}) |DET[DF]|d{ overrightarrow {x}}$
I need to prove that reversing the orientation of some path $gamma$ has the effect of changing the sign of the path integral $int_{gamma}alpha$ for some 1-Form $alpha$.
I think that I need to make a change of coordinates $ overrightarrow {u} = -overrightarrow {u}$, but from here I'm stuck.
Is this the right way to go about proving this? Or is there a better way?
calculus integration multivariable-calculus vector-fields chain-rule
asked Nov 18 at 6:19
Jackson Joffe
335
I need to use the Change of Variables Theorem (basically the Chain Rule), which states that:
Given a change of coordinates $F: mathbb{R}^n to mathbb{R}^n$ of the form $overrightarrow {u} = F ( overrightarrow{x} )$, the integral of $h$ over some region $F(R)$ converts as:
$int_{F(R)} h( overrightarrow {u})d{overrightarrow {u}} = int_Rh(F( overrightarrow {x}) |DET[DF]|d{ overrightarrow {x}}$
I need to prove that reversing the orientation of some path $gamma$ has the effect of changing the sign of the path integral $int_{gamma}alpha$ for some 1-Form $alpha$.
I think that I need to make a change of coordinates $ overrightarrow {u} = -overrightarrow {u}$, but from here I'm stuck.
Is this the right way to go about proving this? Or is there a better way?
calculus integration multivariable-calculus vector-fields chain-rule
calculus integration multivariable-calculus vector-fields chain-rule
asked Nov 18 at 6:19
Jackson Joffe
335
asked Nov 18 at 6:19
Jackson Joffe
335
asked Nov 18 at 6:19
Jackson Joffe
335
asked Nov 18 at 6:19
Jackson Joffe
335
asked Nov 18 at 6:19
Jackson Joffe
335
335
2
Let $gamma(a),gamma(b)$ be the start and endpoint of $gamma$ then consider $gamma(a+b-t)$ over [a,b] for the reverse path
– Fakemistake
Nov 18 at 6:27
add a comment |
2
Let $gamma(a),gamma(b)$ be the start and endpoint of $gamma$ then consider $gamma(a+b-t)$ over [a,b] for the reverse path
– Fakemistake
Nov 18 at 6:27
2
2
Let $gamma(a),gamma(b)$ be the start and endpoint of $gamma$ then consider $gamma(a+b-t)$ over [a,b] for the reverse path
– Fakemistake
Nov 18 at 6:27
Let $gamma(a),gamma(b)$ be the start and endpoint of $gamma$ then consider $gamma(a+b-t)$ over [a,b] for the reverse path
– Fakemistake
Nov 18 at 6:27
add a comment |
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Let $gamma(a),gamma(b)$ be the start and endpoint of $gamma$ then consider $gamma(a+b-t)$ over [a,b] for the reverse path
– Fakemistake
Nov 18 at 6:27