Jacobian matrix
E697756
The Jacobian matrix is a matrix of all first-order partial derivatives of a vector-valued function, fundamental in multivariable calculus for describing how the function locally transforms space.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jacobian matrix canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871789 — 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 matrix Context triple: [Jacobi last multiplier, relatedTo, Jacobian matrix]
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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.
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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.
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C.
Jacobi
Jacobi is a German surname most famously associated with the 19th-century mathematician Carl Gustav Jacob Jacobi, known for his foundational work in elliptic functions and number theory.
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D.
Jacobi operator
The Jacobi operator is a linear differential operator central to the theory of elliptic functions and integrable systems, named after the mathematician Carl Gustav Jacob Jacobi.
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E.
Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots 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 matrix Target entity description: The Jacobian matrix is a matrix of all first-order partial derivatives of a vector-valued function, fundamental in multivariable calculus for describing how the function locally transforms space.
-
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.
Jacobi
Jacobi is a German surname most famously associated with the 19th-century mathematician Carl Gustav Jacob Jacobi, known for his foundational work in elliptic functions and number theory.
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D.
Jacobi operator
The Jacobi operator is a linear differential operator central to the theory of elliptic functions and integrable systems, named after the mathematician Carl Gustav Jacob Jacobi.
-
E.
Sylvester matrix
The Sylvester matrix is a structured matrix constructed from the coefficients of two polynomials, commonly used to compute their resultant and study common roots in algebra.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
matrix ⓘ object in multivariable calculus ⓘ |
| codomainDimension | m ⓘ |
| componentType | first-order partial derivative ⓘ |
| condition | exists when all first-order partial derivatives exist and are continuous near the point ⓘ |
| definedFor |
function from R^n to R^m
ⓘ
vector-valued function ⓘ |
| describes |
how a function locally transforms space
ⓘ
local linear approximation of a function ⓘ |
| domainDimension | n ⓘ |
| entryDefinition | (i,j)-entry is ∂f_i/∂x_j ⓘ |
| field |
computer vision
ⓘ
control theory ⓘ differential geometry ⓘ dynamical systems ⓘ econometrics ⓘ engineering ⓘ machine learning ⓘ mathematics ⓘ multivariable calculus ⓘ numerical analysis ⓘ optimization ⓘ robotics ⓘ vector calculus ⓘ |
| generalizes |
derivative of a scalar function of one variable
ⓘ
gradient ⓘ |
| namedAfter | Carl Gustav Jacob Jacobi NERFINISHED ⓘ |
| property |
depends on choice of coordinates
ⓘ
is linear in the increment of the input ⓘ |
| relatedConcept |
Hessian matrix
NERFINISHED
ⓘ
Jacobian determinant ⓘ differential of a map ⓘ |
| shape | m×n matrix ⓘ |
| specialCaseOf | derivative of a differentiable map between Euclidean spaces ⓘ |
| symbol |
Df
ⓘ
J ⓘ ∂f/∂x ⓘ |
| usedFor |
Newton's method for systems of equations
NERFINISHED
ⓘ
backpropagation in neural networks ⓘ change of variables in multiple integrals ⓘ computing gradients of vector outputs ⓘ coordinate transformations ⓘ differential kinematics in robotics ⓘ error propagation ⓘ implicit function theorem ⓘ inverse function theorem NERFINISHED ⓘ nonlinear system linearization ⓘ sensitivity analysis ⓘ stability analysis of equilibria ⓘ |
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 matrix Description of subject: The Jacobian matrix is a matrix of all first-order partial derivatives of a vector-valued function, fundamental in multivariable calculus for describing how the function locally transforms space.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.