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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Jacobi determinant | 2 |
| Jacobian determinant canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871554 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Jacobian determinant Context triple: [Carl Gustav Jacob Jacobi, knownFor, Jacobian determinant]
-
A.
Jacobi matrix
A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
-
B.
Jacobi bracket
The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
-
C.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
D.
Jacobi last multiplier
The Jacobi last multiplier is a mathematical tool introduced by Carl Gustav Jacob Jacobi for integrating systems of differential equations by providing an integrating factor that simplifies them to solvable form.
-
E.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Jacobian determinant Target entity description: 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.
-
A.
Jacobi matrix
A Jacobi matrix is a tridiagonal matrix, often symmetric, that arises in numerical analysis and mathematical physics, particularly in the study of orthogonal polynomials and eigenvalue problems.
-
B.
Jacobi bracket
The Jacobi bracket is a bilinear operation generalizing the Poisson bracket in differential geometry, central to the theory of Jacobi manifolds and Hamiltonian systems.
-
C.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
D.
Jacobi last multiplier
The Jacobi last multiplier is a mathematical tool introduced by Carl Gustav Jacob Jacobi for integrating systems of differential equations by providing an integrating factor that simplifies them to solvable form.
-
E.
Sylvester determinant
The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.
- F. None of above. chosen
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 ⓘ |
| 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 ⓘ inverse function theorem ⓘ 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Jacobian determinant Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Carl
linked to: Jacobian determinant
subject linked to:
Carl Gustav Jacob
linked to: Jacobian determinant