Help with change of variables in an integral with partial derivatives

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I have a function defined as

$$barw = w(x,y,z)expleft[ int_-infty^z mathrmdz'' psi(x,y,z'')right] tag1$$

and its partial derivative w.r.t $z$ is defined as

$$ fracpartial barwpartial z = A(x,y,z)expleft[ int_-infty^z mathrmdz'' psi(x,y,z'')right] tag2$$

The author has then written

$$w = expleft[- int_-infty^z mathrmdz'' psi(x,y,z'')right] int_-infty^z mathrmdz' A(x,y,z')expleft[ int_-infty^z' mathrmdz'' psi(x,y,z'')right] tag3\
= int_-infty^z mathrmdz' A(x,y,z')expleft[ -int_z'^z mathrmdz'' psi(x,y,z'')right] \$$

I did not understand how the author got equation $(3)$ from equation $(2)$. If an integration of equation $(2)$ has been done w.r.t $z$ how did $z'$ come on the right hand side? and if the integration is done w.r.t $z'$ how did $intfracpartial barwpartial z mathrmdz'$ become $w$?. There are some variable changes that is not obvious to me can anyone explain this?.

integration definite-integrals partial-derivative

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edited Jul 17 at 8:24

asked Jul 16 at 21:12

jsid

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down vote

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I have a function defined as

$$barw = w(x,y,z)expleft[ int_-infty^z mathrmdz'' psi(x,y,z'')right] tag1$$

and its partial derivative w.r.t $z$ is defined as

$$ fracpartial barwpartial z = A(x,y,z)expleft[ int_-infty^z mathrmdz'' psi(x,y,z'')right] tag2$$

The author has then written

$$w = expleft[- int_-infty^z mathrmdz'' psi(x,y,z'')right] int_-infty^z mathrmdz' A(x,y,z')expleft[ int_-infty^z' mathrmdz'' psi(x,y,z'')right] tag3\
= int_-infty^z mathrmdz' A(x,y,z')expleft[ -int_z'^z mathrmdz'' psi(x,y,z'')right] \$$

I did not understand how the author got equation $(3)$ from equation $(2)$. If an integration of equation $(2)$ has been done w.r.t $z$ how did $z'$ come on the right hand side? and if the integration is done w.r.t $z'$ how did $intfracpartial barwpartial z mathrmdz'$ become $w$?. There are some variable changes that is not obvious to me can anyone explain this?.

integration definite-integrals partial-derivative

share|cite|improve this question

edited Jul 17 at 8:24

asked Jul 16 at 21:12

jsid

12

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I have a function defined as

$$barw = w(x,y,z)expleft[ int_-infty^z mathrmdz'' psi(x,y,z'')right] tag1$$

and its partial derivative w.r.t $z$ is defined as

$$ fracpartial barwpartial z = A(x,y,z)expleft[ int_-infty^z mathrmdz'' psi(x,y,z'')right] tag2$$

The author has then written

$$w = expleft[- int_-infty^z mathrmdz'' psi(x,y,z'')right] int_-infty^z mathrmdz' A(x,y,z')expleft[ int_-infty^z' mathrmdz'' psi(x,y,z'')right] tag3\
= int_-infty^z mathrmdz' A(x,y,z')expleft[ -int_z'^z mathrmdz'' psi(x,y,z'')right] \$$

I did not understand how the author got equation $(3)$ from equation $(2)$. If an integration of equation $(2)$ has been done w.r.t $z$ how did $z'$ come on the right hand side? and if the integration is done w.r.t $z'$ how did $intfracpartial barwpartial z mathrmdz'$ become $w$?. There are some variable changes that is not obvious to me can anyone explain this?.

integration definite-integrals partial-derivative

share|cite|improve this question

edited Jul 17 at 8:24

asked Jul 16 at 21:12

jsid

12

I have a function defined as

$$barw = w(x,y,z)expleft[ int_-infty^z mathrmdz'' psi(x,y,z'')right] tag1$$

and its partial derivative w.r.t $z$ is defined as

$$ fracpartial barwpartial z = A(x,y,z)expleft[ int_-infty^z mathrmdz'' psi(x,y,z'')right] tag2$$

The author has then written

$$w = expleft[- int_-infty^z mathrmdz'' psi(x,y,z'')right] int_-infty^z mathrmdz' A(x,y,z')expleft[ int_-infty^z' mathrmdz'' psi(x,y,z'')right] tag3\
= int_-infty^z mathrmdz' A(x,y,z')expleft[ -int_z'^z mathrmdz'' psi(x,y,z'')right] \$$

I did not understand how the author got equation $(3)$ from equation $(2)$. If an integration of equation $(2)$ has been done w.r.t $z$ how did $z'$ come on the right hand side? and if the integration is done w.r.t $z'$ how did $intfracpartial barwpartial z mathrmdz'$ become $w$?. There are some variable changes that is not obvious to me can anyone explain this?.

integration definite-integrals partial-derivative

share|cite|improve this question

edited Jul 17 at 8:24

asked Jul 16 at 21:12

jsid

12

share|cite|improve this question

share|cite|improve this question

edited Jul 17 at 8:24

edited Jul 17 at 8:24

edited Jul 17 at 8:24

asked Jul 16 at 21:12

jsid

12

asked Jul 16 at 21:12

jsid

12

asked Jul 16 at 21:12

jsid

12

12

add a comment | 

add a comment | 

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