Line integral in proof of Green's theorem
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In wikipedia page about Green's theorem the following equality appears:
$$
int_{C_1} L(x,y), dx = int_a^b L(x,g_1(x)), dx
$$
I do not understand it. Wikipedia page about line integral defines line integral, when applied to an scalar function, as:
$$
int_{mathcal{C}} f(mathbf{r}), ds = int_a^b fleft(mathbf{r}(t)right)|mathbf{r}'(t)| , dt.
$$
that, applied to the proof expression and taken into account that the curve $C_1$ has been parametrized as $(x,g_1(x))$, gives (?):
$$
int_{C_1} L(x,y), dx = int_a^b L(x,g_1(x)) ,, |(1,g_1'(x)| ,, dx
$$
that seems different to the one said in the proof (all curve derivative term has been supresed).
calculus line-integrals greens-theorem
asked Dec 22 '18 at 18:54
pasaba por aquipasaba por aqui
454316
$endgroup$
add a comment |
$begingroup$
In wikipedia page about Green's theorem the following equality appears:
$$
int_{C_1} L(x,y), dx = int_a^b L(x,g_1(x)), dx
$$
I do not understand it. Wikipedia page about line integral defines line integral, when applied to an scalar function, as:
$$
int_{mathcal{C}} f(mathbf{r}), ds = int_a^b fleft(mathbf{r}(t)right)|mathbf{r}'(t)| , dt.
$$
that, applied to the proof expression and taken into account that the curve $C_1$ has been parametrized as $(x,g_1(x))$, gives (?):
$$
int_{C_1} L(x,y), dx = int_a^b L(x,g_1(x)) ,, |(1,g_1'(x)| ,, dx
$$
that seems different to the one said in the proof (all curve derivative term has been supresed).
calculus line-integrals greens-theorem
asked Dec 22 '18 at 18:54
pasaba por aquipasaba por aqui
454316
$endgroup$
add a comment |
$begingroup$
In wikipedia page about Green's theorem the following equality appears:
$$
int_{C_1} L(x,y), dx = int_a^b L(x,g_1(x)), dx
$$
I do not understand it. Wikipedia page about line integral defines line integral, when applied to an scalar function, as:
$$
int_{mathcal{C}} f(mathbf{r}), ds = int_a^b fleft(mathbf{r}(t)right)|mathbf{r}'(t)| , dt.
$$
that, applied to the proof expression and taken into account that the curve $C_1$ has been parametrized as $(x,g_1(x))$, gives (?):
$$
int_{C_1} L(x,y), dx = int_a^b L(x,g_1(x)) ,, |(1,g_1'(x)| ,, dx
$$
that seems different to the one said in the proof (all curve derivative term has been supresed).
calculus line-integrals greens-theorem
asked Dec 22 '18 at 18:54
pasaba por aquipasaba por aqui
454316
$endgroup$
In wikipedia page about Green's theorem the following equality appears:
$$
int_{C_1} L(x,y), dx = int_a^b L(x,g_1(x)), dx
$$
I do not understand it. Wikipedia page about line integral defines line integral, when applied to an scalar function, as:
$$
int_{mathcal{C}} f(mathbf{r}), ds = int_a^b fleft(mathbf{r}(t)right)|mathbf{r}'(t)| , dt.
$$
that, applied to the proof expression and taken into account that the curve $C_1$ has been parametrized as $(x,g_1(x))$, gives (?):
$$
int_{C_1} L(x,y), dx = int_a^b L(x,g_1(x)) ,, |(1,g_1'(x)| ,, dx
$$
that seems different to the one said in the proof (all curve derivative term has been supresed).
calculus line-integrals greens-theorem
calculus line-integrals greens-theorem
asked Dec 22 '18 at 18:54
pasaba por aquipasaba por aqui
454316
asked Dec 22 '18 at 18:54
pasaba por aquipasaba por aqui
454316
asked Dec 22 '18 at 18:54
pasaba por aquipasaba por aqui
454316
asked Dec 22 '18 at 18:54
pasaba por aquipasaba por aqui
454316
asked Dec 22 '18 at 18:54
pasaba por aquipasaba por aqui
454316
454316
add a comment |
add a comment |
1 Answer
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Note that one integral is a $ds$ integral and the other integral is a $dx$ integral. Here, Green's Theorem is written in the $int L,dx+M,dy$ form.
answered Dec 22 '18 at 18:56
Ted ShifrinTed Shifrin
64.6k44692
$endgroup$
$begingroup$
Thanks for your answer. What is the dx integral of an scalar function over a curve $C_1$ ?
$endgroup$
– pasaba por aqui
Dec 22 '18 at 19:00
1
$begingroup$
@pasabaporaqui Look at tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx Contrast it with the previous part on line integrals with respect to arc length.
$endgroup$
– saulspatz
Dec 22 '18 at 19:59
$begingroup$
@saulspatz: thanks a lot
$endgroup$
– pasaba por aqui
Dec 22 '18 at 21:51
add a comment |
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$begingroup$
Note that one integral is a $ds$ integral and the other integral is a $dx$ integral. Here, Green's Theorem is written in the $int L,dx+M,dy$ form.
answered Dec 22 '18 at 18:56
Ted ShifrinTed Shifrin
64.6k44692
$endgroup$
$begingroup$
Thanks for your answer. What is the dx integral of an scalar function over a curve $C_1$ ?
$endgroup$
– pasaba por aqui
Dec 22 '18 at 19:00
1
$begingroup$
@pasabaporaqui Look at tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx Contrast it with the previous part on line integrals with respect to arc length.
$endgroup$
– saulspatz
Dec 22 '18 at 19:59
$begingroup$
@saulspatz: thanks a lot
$endgroup$
– pasaba por aqui
Dec 22 '18 at 21:51
add a comment |
$begingroup$
Note that one integral is a $ds$ integral and the other integral is a $dx$ integral. Here, Green's Theorem is written in the $int L,dx+M,dy$ form.
answered Dec 22 '18 at 18:56
Ted ShifrinTed Shifrin
64.6k44692
$endgroup$
$begingroup$
Thanks for your answer. What is the dx integral of an scalar function over a curve $C_1$ ?
$endgroup$
– pasaba por aqui
Dec 22 '18 at 19:00
1
$begingroup$
@pasabaporaqui Look at tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx Contrast it with the previous part on line integrals with respect to arc length.
$endgroup$
– saulspatz
Dec 22 '18 at 19:59
$begingroup$
@saulspatz: thanks a lot
$endgroup$
– pasaba por aqui
Dec 22 '18 at 21:51
add a comment |
$begingroup$
Note that one integral is a $ds$ integral and the other integral is a $dx$ integral. Here, Green's Theorem is written in the $int L,dx+M,dy$ form.
answered Dec 22 '18 at 18:56
Ted ShifrinTed Shifrin
64.6k44692
$endgroup$
Note that one integral is a $ds$ integral and the other integral is a $dx$ integral. Here, Green's Theorem is written in the $int L,dx+M,dy$ form.
answered Dec 22 '18 at 18:56
Ted ShifrinTed Shifrin
64.6k44692
answered Dec 22 '18 at 18:56
Ted ShifrinTed Shifrin
64.6k44692
answered Dec 22 '18 at 18:56
Ted ShifrinTed Shifrin
64.6k44692
answered Dec 22 '18 at 18:56
Ted ShifrinTed Shifrin
64.6k44692
64.6k44692
$begingroup$
Thanks for your answer. What is the dx integral of an scalar function over a curve $C_1$ ?
$endgroup$
– pasaba por aqui
Dec 22 '18 at 19:00
1
$begingroup$
@pasabaporaqui Look at tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx Contrast it with the previous part on line integrals with respect to arc length.
$endgroup$
– saulspatz
Dec 22 '18 at 19:59
$begingroup$
@saulspatz: thanks a lot
$endgroup$
– pasaba por aqui
Dec 22 '18 at 21:51
add a comment |
$begingroup$
Thanks for your answer. What is the dx integral of an scalar function over a curve $C_1$ ?
$endgroup$
– pasaba por aqui
Dec 22 '18 at 19:00
1
$begingroup$
@pasabaporaqui Look at tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx Contrast it with the previous part on line integrals with respect to arc length.
$endgroup$
– saulspatz
Dec 22 '18 at 19:59
$begingroup$
@saulspatz: thanks a lot
$endgroup$
– pasaba por aqui
Dec 22 '18 at 21:51
$begingroup$
Thanks for your answer. What is the dx integral of an scalar function over a curve $C_1$ ?
$endgroup$
– pasaba por aqui
Dec 22 '18 at 19:00
$begingroup$
Thanks for your answer. What is the dx integral of an scalar function over a curve $C_1$ ?
$endgroup$
– pasaba por aqui
Dec 22 '18 at 19:00
1
1
$begingroup$
@pasabaporaqui Look at tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx Contrast it with the previous part on line integrals with respect to arc length.
$endgroup$
– saulspatz
Dec 22 '18 at 19:59
$begingroup$
@pasabaporaqui Look at tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx Contrast it with the previous part on line integrals with respect to arc length.
$endgroup$
– saulspatz
Dec 22 '18 at 19:59
$begingroup$
@saulspatz: thanks a lot
$endgroup$
– pasaba por aqui
Dec 22 '18 at 21:51
$begingroup$
@saulspatz: thanks a lot
$endgroup$
– pasaba por aqui
Dec 22 '18 at 21:51
add a comment |
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