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When we are taught about multiple integrals we are taught about change of variables and the resulting Jacobian that accounts for the change in area/vol element as you move from one coord. system to another. Assuming that one performs a linear change of variables, can these coordinate transformations be thought of as change of basis matrix? That is, can I view $P^-1AP = B$ as taking the area element (columns of $A$) into the area element in the new coord. system (columns of $B$), where $P$ are the linear change of variables. If so, then since det is invariant under this operation, why do we have to worry about a change in area/vol of the area/vol element?

How does change of basis and change of coordinates relate (if at all) to each other and how do you interpret area being invariant in one situation and not the other?

linear-algebra multivariable-calculus linear-transformations determinant jacobian

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asked Mar 13 at 18:04

jeffjeff

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1

1

$begingroup$

When we are taught about multiple integrals we are taught about change of variables and the resulting Jacobian that accounts for the change in area/vol element as you move from one coord. system to another. Assuming that one performs a linear change of variables, can these coordinate transformations be thought of as change of basis matrix? That is, can I view $P^-1AP = B$ as taking the area element (columns of $A$) into the area element in the new coord. system (columns of $B$), where $P$ are the linear change of variables. If so, then since det is invariant under this operation, why do we have to worry about a change in area/vol of the area/vol element?

How does change of basis and change of coordinates relate (if at all) to each other and how do you interpret area being invariant in one situation and not the other?

linear-algebra multivariable-calculus linear-transformations determinant jacobian

share|cite|improve this question

asked Mar 13 at 18:04

jeffjeff

61

New contributor

jeff is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

add a comment | 

$begingroup$

When we are taught about multiple integrals we are taught about change of variables and the resulting Jacobian that accounts for the change in area/vol element as you move from one coord. system to another. Assuming that one performs a linear change of variables, can these coordinate transformations be thought of as change of basis matrix? That is, can I view $P^-1AP = B$ as taking the area element (columns of $A$) into the area element in the new coord. system (columns of $B$), where $P$ are the linear change of variables. If so, then since det is invariant under this operation, why do we have to worry about a change in area/vol of the area/vol element?

How does change of basis and change of coordinates relate (if at all) to each other and how do you interpret area being invariant in one situation and not the other?

linear-algebra multivariable-calculus linear-transformations determinant jacobian

share|cite|improve this question

asked Mar 13 at 18:04

jeffjeff

61

New contributor

jeff is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

asked Mar 13 at 18:04

jeffjeff

61

New contributor

jeff is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

asked Mar 13 at 18:04

jeffjeff

61

New contributor

jeff is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 18:04

jeffjeff

61

New contributor

jeff is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Mar 13 at 18:04

jeffjeff

61