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.
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 ⓘ |
Referenced by (2)
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