How to prove one theorem related to Stoke`s theorem
$begingroup$
Stock`s theorem $$ointlimits_C {{bf{a}} cdot {bf{dr}}} = iintlimits_S {nabla times {bf{a}}, cdot {bf{n}}dA}$$
Substituting ${bf{a}} = {bf{f}} times {bf{c}}$
we find that $$ointlimits_C {{bf{dr}} times {bf{f}}} = iintlimits_S {left( {{bf{n}} times nabla ,} right) times {bf{f}}dA}$$
since
${bf{n}} cdot left( {nabla times left( {{bf{f}} times {bf{c}}} right)} right) = {bf{c}} cdot left( {left( {{bf{n}} timesnabla } right) times {bf{f}}} right)
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfi
% fHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk
% 0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9
% Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHUbGaey
% yXIC9aaeWaaeaadaWhcaqaaiabgEGirdGaay51GaGaey41aq7aaeWa
% aeaacaWHMbGaey41aqRaaC4yaaGaayjkaiaawMcaaaGaayjkaiaawM
% caaiabg2da9iaahogacqGHflY1daqadaqaamaabmaabaGaaCOBaiab
% gEna0oaaFiaabaGaey4bIenacaGLxdcaaiaawIcacaGLPaaacqGHxd
% aTcaWHMbaacaGLOaGaayzkaaaaaa!55F5!
$
I can`t prove last equation, please help,
c is constant vector,
n is unit vector,
a and f are vector functions
curl
asked Dec 17 '18 at 10:56
SergeyFominSergeyFomin
1187
$endgroup$
add a comment |
$begingroup$
Stock`s theorem $$ointlimits_C {{bf{a}} cdot {bf{dr}}} = iintlimits_S {nabla times {bf{a}}, cdot {bf{n}}dA}$$
Substituting ${bf{a}} = {bf{f}} times {bf{c}}$
we find that $$ointlimits_C {{bf{dr}} times {bf{f}}} = iintlimits_S {left( {{bf{n}} times nabla ,} right) times {bf{f}}dA}$$
since
${bf{n}} cdot left( {nabla times left( {{bf{f}} times {bf{c}}} right)} right) = {bf{c}} cdot left( {left( {{bf{n}} timesnabla } right) times {bf{f}}} right)
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfi
% fHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk
% 0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9
% Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHUbGaey
% yXIC9aaeWaaeaadaWhcaqaaiabgEGirdGaay51GaGaey41aq7aaeWa
% aeaacaWHMbGaey41aqRaaC4yaaGaayjkaiaawMcaaaGaayjkaiaawM
% caaiabg2da9iaahogacqGHflY1daqadaqaamaabmaabaGaaCOBaiab
% gEna0oaaFiaabaGaey4bIenacaGLxdcaaiaawIcacaGLPaaacqGHxd
% aTcaWHMbaacaGLOaGaayzkaaaaaa!55F5!
$
I can`t prove last equation, please help,
c is constant vector,
n is unit vector,
a and f are vector functions
curl
asked Dec 17 '18 at 10:56
SergeyFominSergeyFomin
1187
$endgroup$
$begingroup$
$f$ is a function? $n$ a unit vector? What is $ntimesnabla$? In any case it appears be a trivial (but long) calculation.
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 17 '18 at 17:47
$begingroup$
Yes, f is vector function, n is unit vector, $${bf{n}} times nabla % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOBaiabgE % na0kabgEGirdaa!3A8A! $$ is a cross product.
$endgroup$
– SergeyFomin
Dec 18 '18 at 20:41
$begingroup$
And what is the "cross product" of vector $times$ operator?
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 19 '18 at 15:03
$begingroup$
It`s a vector operator ${bf{n}} times nabla = {n_i}{{{bf{hat e}}}_i} times {{{bf{hat e}}}_j}{partial _j} = {{{bf{hat e}}}_k}{varepsilon _{ijk}}{n_i}{partial _j}$
$endgroup$
– SergeyFomin
Dec 20 '18 at 8:09
add a comment |
$begingroup$
Stock`s theorem $$ointlimits_C {{bf{a}} cdot {bf{dr}}} = iintlimits_S {nabla times {bf{a}}, cdot {bf{n}}dA}$$
Substituting ${bf{a}} = {bf{f}} times {bf{c}}$
we find that $$ointlimits_C {{bf{dr}} times {bf{f}}} = iintlimits_S {left( {{bf{n}} times nabla ,} right) times {bf{f}}dA}$$
since
${bf{n}} cdot left( {nabla times left( {{bf{f}} times {bf{c}}} right)} right) = {bf{c}} cdot left( {left( {{bf{n}} timesnabla } right) times {bf{f}}} right)
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfi
% fHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk
% 0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9
% Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHUbGaey
% yXIC9aaeWaaeaadaWhcaqaaiabgEGirdGaay51GaGaey41aq7aaeWa
% aeaacaWHMbGaey41aqRaaC4yaaGaayjkaiaawMcaaaGaayjkaiaawM
% caaiabg2da9iaahogacqGHflY1daqadaqaamaabmaabaGaaCOBaiab
% gEna0oaaFiaabaGaey4bIenacaGLxdcaaiaawIcacaGLPaaacqGHxd
% aTcaWHMbaacaGLOaGaayzkaaaaaa!55F5!
$
I can`t prove last equation, please help,
c is constant vector,
n is unit vector,
a and f are vector functions
curl
asked Dec 17 '18 at 10:56
SergeyFominSergeyFomin
1187
$endgroup$
Stock`s theorem $$ointlimits_C {{bf{a}} cdot {bf{dr}}} = iintlimits_S {nabla times {bf{a}}, cdot {bf{n}}dA}$$
Substituting ${bf{a}} = {bf{f}} times {bf{c}}$
we find that $$ointlimits_C {{bf{dr}} times {bf{f}}} = iintlimits_S {left( {{bf{n}} times nabla ,} right) times {bf{f}}dA}$$
since
${bf{n}} cdot left( {nabla times left( {{bf{f}} times {bf{c}}} right)} right) = {bf{c}} cdot left( {left( {{bf{n}} timesnabla } right) times {bf{f}}} right)
% MathType!MTEF!2!1!+-
% feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYfdmGievaebbnrfi
% fHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk
% 0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9
% Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHUbGaey
% yXIC9aaeWaaeaadaWhcaqaaiabgEGirdGaay51GaGaey41aq7aaeWa
% aeaacaWHMbGaey41aqRaaC4yaaGaayjkaiaawMcaaaGaayjkaiaawM
% caaiabg2da9iaahogacqGHflY1daqadaqaamaabmaabaGaaCOBaiab
% gEna0oaaFiaabaGaey4bIenacaGLxdcaaiaawIcacaGLPaaacqGHxd
% aTcaWHMbaacaGLOaGaayzkaaaaaa!55F5!
$
I can`t prove last equation, please help,
c is constant vector,
n is unit vector,
a and f are vector functions
curl
curl
asked Dec 17 '18 at 10:56
SergeyFominSergeyFomin
1187
asked Dec 17 '18 at 10:56
SergeyFominSergeyFomin
1187
edited Dec 19 '18 at 6:33
SergeyFomin
asked Dec 17 '18 at 10:56
SergeyFominSergeyFomin
1187
asked Dec 17 '18 at 10:56
SergeyFominSergeyFomin
1187
asked Dec 17 '18 at 10:56
SergeyFominSergeyFomin
1187
1187
$begingroup$
$f$ is a function? $n$ a unit vector? What is $ntimesnabla$? In any case it appears be a trivial (but long) calculation.
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 17 '18 at 17:47
$begingroup$
Yes, f is vector function, n is unit vector, $${bf{n}} times nabla % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOBaiabgE % na0kabgEGirdaa!3A8A! $$ is a cross product.
$endgroup$
– SergeyFomin
Dec 18 '18 at 20:41
$begingroup$
And what is the "cross product" of vector $times$ operator?
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 19 '18 at 15:03
$begingroup$
It`s a vector operator ${bf{n}} times nabla = {n_i}{{{bf{hat e}}}_i} times {{{bf{hat e}}}_j}{partial _j} = {{{bf{hat e}}}_k}{varepsilon _{ijk}}{n_i}{partial _j}$
$endgroup$
– SergeyFomin
Dec 20 '18 at 8:09
add a comment |
$begingroup$
$f$ is a function? $n$ a unit vector? What is $ntimesnabla$? In any case it appears be a trivial (but long) calculation.
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 17 '18 at 17:47
$begingroup$
Yes, f is vector function, n is unit vector, $${bf{n}} times nabla % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOBaiabgE % na0kabgEGirdaa!3A8A! $$ is a cross product.
$endgroup$
– SergeyFomin
Dec 18 '18 at 20:41
$begingroup$
And what is the "cross product" of vector $times$ operator?
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 19 '18 at 15:03
$begingroup$
It`s a vector operator ${bf{n}} times nabla = {n_i}{{{bf{hat e}}}_i} times {{{bf{hat e}}}_j}{partial _j} = {{{bf{hat e}}}_k}{varepsilon _{ijk}}{n_i}{partial _j}$
$endgroup$
– SergeyFomin
Dec 20 '18 at 8:09
$begingroup$
$f$ is a function? $n$ a unit vector? What is $ntimesnabla$? In any case it appears be a trivial (but long) calculation.
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 17 '18 at 17:47
$begingroup$
$f$ is a function? $n$ a unit vector? What is $ntimesnabla$? In any case it appears be a trivial (but long) calculation.
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 17 '18 at 17:47
$begingroup$
Yes, f is vector function, n is unit vector, $${bf{n}} times nabla % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOBaiabgE % na0kabgEGirdaa!3A8A! $$ is a cross product.
$endgroup$
– SergeyFomin
Dec 18 '18 at 20:41
$begingroup$
Yes, f is vector function, n is unit vector, $${bf{n}} times nabla % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOBaiabgE % na0kabgEGirdaa!3A8A! $$ is a cross product.
$endgroup$
– SergeyFomin
Dec 18 '18 at 20:41
$begingroup$
And what is the "cross product" of vector $times$ operator?
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 19 '18 at 15:03
$begingroup$
And what is the "cross product" of vector $times$ operator?
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 19 '18 at 15:03
$begingroup$
It`s a vector operator ${bf{n}} times nabla = {n_i}{{{bf{hat e}}}_i} times {{{bf{hat e}}}_j}{partial _j} = {{{bf{hat e}}}_k}{varepsilon _{ijk}}{n_i}{partial _j}$
$endgroup$
– SergeyFomin
Dec 20 '18 at 8:09
$begingroup$
It`s a vector operator ${bf{n}} times nabla = {n_i}{{{bf{hat e}}}_i} times {{{bf{hat e}}}_j}{partial _j} = {{{bf{hat e}}}_k}{varepsilon _{ijk}}{n_i}{partial _j}$
$endgroup$
– SergeyFomin
Dec 20 '18 at 8:09
add a comment |
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$begingroup$
$f$ is a function? $n$ a unit vector? What is $ntimesnabla$? In any case it appears be a trivial (but long) calculation.
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 17 '18 at 17:47
$begingroup$
Yes, f is vector function, n is unit vector, $${bf{n}} times nabla % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCOBaiabgE % na0kabgEGirdaa!3A8A! $$ is a cross product.
$endgroup$
– SergeyFomin
Dec 18 '18 at 20:41
$begingroup$
And what is the "cross product" of vector $times$ operator?
$endgroup$
– Martín-Blas Pérez Pinilla
Dec 19 '18 at 15:03
$begingroup$
It`s a vector operator ${bf{n}} times nabla = {n_i}{{{bf{hat e}}}_i} times {{{bf{hat e}}}_j}{partial _j} = {{{bf{hat e}}}_k}{varepsilon _{ijk}}{n_i}{partial _j}$
$endgroup$
– SergeyFomin
Dec 20 '18 at 8:09