Jacobian determinant
E697751
The Jacobian determinant is a scalar value derived from the Jacobian matrix that measures how a multivariable function locally scales and distorts volume under a change of variables.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Jacobi determinant | 1 |
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
determinant
ⓘ
mathematical concept ⓘ scalar quantity ⓘ tool in multivariable calculus ⓘ |
| appliesTo |
coordinate charts on manifolds
ⓘ
differentiable functions between Euclidean spaces ⓘ maps from R^n to R^n ⓘ |
| characterizes | local behavior of differentiable maps ⓘ |
| context |
coordinate changes in physics
ⓘ
geometric measure theory ⓘ n-dimensional integration ⓘ transformations in statistics ⓘ |
| definedFrom | Jacobian matrix ⓘ |
| describes |
local area scaling
ⓘ
local orientation change ⓘ local volume scaling ⓘ |
| field |
analysis
ⓘ
differential geometry ⓘ measure theory ⓘ multivariable calculus ⓘ vector calculus ⓘ |
| generalizes | one-dimensional derivative as scaling factor ⓘ |
| hasInput | Jacobian matrix ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi NERFINISHED ⓘ |
| outputType | real number ⓘ |
| property |
absolute value gives local volume scaling factor
ⓘ
can be positive or negative ⓘ depends on point in domain ⓘ equals determinant of matrix of first partial derivatives ⓘ nonzero value implies local invertibility ⓘ sign encodes orientation preservation or reversal ⓘ zero value indicates local non-invertibility ⓘ |
| relatedTo |
change of variables formula
ⓘ
determinant of linear transformations ⓘ differential forms ⓘ implicit function theorem NERFINISHED ⓘ inverse function theorem NERFINISHED ⓘ linear approximation of maps ⓘ orientation of manifolds ⓘ |
| specialCaseOf | determinant of derivative map ⓘ |
| symbol |
\det\left(\frac{\partial(x_1,\dots,x_n)}{\partial(u_1,\dots,u_n)}\right)
ⓘ
det(J) ⓘ |J| ⓘ |
| usedFor |
change of variables in multiple integrals
ⓘ
computing density transformations ⓘ coordinate transformations ⓘ nonlinear change of variables ⓘ probability density change under transformations ⓘ transforming area elements ⓘ transforming line elements ⓘ transforming volume elements ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.