the Jacobian Matrix of a multivariable vector-valued function 𝑓 is the matrix of all its first-order partial derivatives
is the matrix representing the best linear map approximation of 𝑓 near every point
takes advantage of the fact that most multivariable functions are locally linear

Jacobian Matrix - Definition

The Jacobian Matrix 𝐉 of a multivariable function 𝒇: ℝ^𝑛 → ℝ^𝑚 is defined as an 𝑚×𝑛 matrix whose (𝑖,𝑗)^𝑡𝘩 entry is 𝐉_𝑖𝑗 = 𝛿𝑓_𝑖 / 𝛿𝑥_𝑗

Indent

Jacobian Matrix - Examples

Example 1

𝒇: ℝ² → ℝ³

since 𝐉 is not a square matrix, there is no Jacobian Determinant

Example 2

𝒇: ℝ² → ℝ²

since 𝐉 is a square matrix, the Jacobian Determinant is: 1 - 𝑠𝑖𝑛(𝑥)𝑠𝑖𝑛(𝑦)