Jacobian matrix and determinant
Example 1

Consider the function f : R → R, with (x, y) ↦ (f (x, y), f (x, y)), given by f () = =. {\displaystyle \mathbf {f} \left({\begin{bmatrix}x\\y\end{bmatrix}}\right)={\begin{bmatrix}f_{1}(x,y)\\f_{2}(x,y)\end{bmatrix}}={\begin{bmatrix}x^{2}y\\5x+\sin y\end{bmatrix}}.}

Then we have f 1 (x, y) = x 2 y {\displaystyle f_{1}(x,y)=x^{2}y} and f 2 (x, y) = 5 x + sin y {\displaystyle f_{2}(x,y)=5x+\sin y} and the Jacobian matrix of f is J f (x, y) = = {\displaystyle \mathbf {J} _{\mathbf {f} }(x,y)={\begin{bmatrix}{\dfrac {\partial f_{1}}{\partial x}}&{\dfrac {\partial f_{1}}{\partial y}}\\{\dfrac {\partial f_{2}}{\partial x}}&{\dfrac {\partial f_{2}}{\partial y}}\end{bmatrix}}={\begin{bmatrix}2xy&x^{2}\\5&\cos y\end{bmatrix}}} and the Jacobian determinant is det (J f (x, y)) = 2 x y cos y − 5 x 2. {\displaystyle \det(\mathbf {J} _{\mathbf {f} }(x,y))=2xy\cos y-5x^{2}.}
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