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University: University of California, San Diego - UCSDGreek Organization: Lambda Chi Alpha
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Comment: Suppose f : Rn → Rm is a function such that each of its first-order partial derivatives exist on Rn. This function takes a point x ∈ Rn as input and produces the vector f(x) ∈ Rm as output. Then the Jacobian matrix of f is defined to be an m×n matrix, denoted by J, whose (i,j)th entry is � � � = ∂ � � ∂ � � {\textstyle \mathbf {J} _{ij}={\frac {\partial f_{i}}{\partial x_{j}}}}, or explicitly � = [ ∂ � ∂ � 1 ⋯ ∂ � ∂ � � ] = [ ∇ T � 1 ⋮ ∇ T � � ] = [ ∂ � 1 ∂ � 1 ⋯ ∂ � 1 ∂ � � ⋮ ⋱ ⋮ ∂ � � ∂ � 1 ⋯ ∂ � � ∂ � � ] {\displaystyle \mathbf {J} ={\begin{bmatrix}{\dfrac {\partial \mathbf {f} }{\partial x_{1}}}&\cdots &{\dfrac {\partial \mathbf {f} }{\partial x_{n}}}\end{bmatrix}}={\begin{bmatrix}\nabla ^{\mathrm {T} }f_{1}\\\vdots \\\nabla ^{\mathrm {T} }f_{m}\end{bmatrix}}={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial f_{m}}{\partial x_{1}}}&\cdots &{\dfrac {\partial f_{m}}{\partial x_{n}}}\end{bmatrix}}} where ∇ T � � {\displaystyle \nabla ^{\mathrm {T} }f_{i}} is the transpose (row vector) of the gradient of the � i-th component.
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