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| A '''Jacobian matrix''' is a square matrix of [[Calculus/PartialDerivative|partial derivatives]] describing a [[Calculus/CoordinateSystem#Changing_System|change of coordinate systems]] or, more generally, a [[LinearAlgebra/Basis#Change_of_Basis|change of basis]]. | A '''Jacobian matrix''' is a square matrix of [[Calculus/PartialDerivative|partial derivatives]]. |
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| The [[LinearAlgebra/Determinant|determinant]] of such a matrix is the '''Jacobian determinant'''. | A Jacobian is most commonly used to describe a [[Calculus/CoordinateSystem#Changing_System|change of coordinate systems]] or a [[LinearAlgebra/Basis#Change_of_Basis|change of basis]]. As the result, the [[LinearAlgebra/Determinant|determinant]] of such a matrix is called '''Jacobian determinant'''. |
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| Consider a vector-valued function, such as a [[Calculus/ParametricEquation|parameterized]] function, but for simplicity let ''F(X) = <f,,1,,(X), f,,2,,(X)>'' where ''X'' is a vector as ''X = <x,,1,,``, x,,2,,``, ... x,,n,,>''. A [[Calculus/Gradient|gradient]] can be calculated for each of the component function. {{attachment:jacobian.svg}} === Relation to Hessian Matrices === A [[Calculus/HessianMatrix|Hessian matrix]] is also a square matrix of partial derivatives. Importantly however, it is formed by all possible second-order derivatives of a scalar-valued function. As such, it is more closely related to a gradient than a Jacobian matrix. To express all possible second-order derivatives of a vector-valued function, a tensor would in fact be required. ---- == Usage == === Change of Coordinate Systems === |
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| Generically, consider the transformation from ''(x,y)'' coordinates to ''(u,v)'' coordinates. The Jacobian matrix is given by: {{{ ┌ ┐ | ∂x ∂x | | ―― ―― | | ∂u ∂v | | | | ∂y ∂y | | ―― ―― | | ∂u ∂v | └ ┘ }}} The [[LinearAlgebra/Determinant|determinant]] of any matrix describes its scaling factor in space. For a Jacobian matrix, this is the '''Jacobian determinant'''. {{{ | ∂x ∂x | | ―― ―― | | ∂u ∂v | ∂x ∂y ∂x ∂y det | | = ―― ―― - ―― ―― | ∂y ∂y | ∂u ∂v ∂v ∂u | ―― ―― | | ∂u ∂v | }}} Note that determinants can be positive or negative. A negative Jacobian determinant simply means that space was flipped; the true scaling factor is given by the absolute value of the Jacobian determinant. |
The [[LinearAlgebra/Determinant|determinant]] of any matrix describes its scaling factor in space. For a Jacobian matrix, this is the '''Jacobian determinant'''. Note that determinants can be positive or negative. A negative Jacobian determinant simply means that space was flipped; the true scaling factor is given by the absolute value of the Jacobian determinant. |
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| {{{ | ∂x ∂x | | ―― ―― | | ∂θ ∂r | | cosθ -r*sinθ | det | | = det | sinθ r*cosθ | = (cosθ)(r*cosθ) - (-r*sinθ)(sinθ) = r | ∂y ∂y | | ―― ―― | | ∂θ ∂r | }}} |
{{attachment:polar.svg}} |
Jacobian Matrices and Determinants
A Jacobian matrix is a square matrix of partial derivatives.
A Jacobian is most commonly used to describe a change of coordinate systems or a change of basis. As the result, the determinant of such a matrix is called Jacobian determinant.
Contents
Description
Consider a vector-valued function, such as a parameterized function, but for simplicity let F(X) = <f1(X), f2(X)> where X is a vector as X = <x1, x2, ... xn>. A gradient can be calculated for each of the component function.
Relation to Hessian Matrices
A Hessian matrix is also a square matrix of partial derivatives. Importantly however, it is formed by all possible second-order derivatives of a scalar-valued function. As such, it is more closely related to a gradient than a Jacobian matrix.
To express all possible second-order derivatives of a vector-valued function, a tensor would in fact be required.
Usage
Change of Coordinate Systems
Some differentiation problems are more easily solved in a different coordinate system. The transformation of points is straightforward and known. The trick is that space (i.e., area in 2 dimensions, volume in 3, and so on) was also transformed. For a given transformation, the Jacobian matrix contains all partial derivatives involved.
The determinant of any matrix describes its scaling factor in space. For a Jacobian matrix, this is the Jacobian determinant. Note that determinants can be positive or negative. A negative Jacobian determinant simply means that space was flipped; the true scaling factor is given by the absolute value of the Jacobian determinant.
Consider the transformation from polar to Cartesian coordinates. The Jacobian matrix and determinant are given by:
Therefore dxdy = rdrdθ.
The Jacobian determinant also explains the chain rule.