Calculus Small Changes, Big Ideas

Section 18.2 The Jacobian matrix and its determinant

• The Jacobian matrix of a transformation \(\bv f\) given by \(x=g(u,v)\text{,}\) \(y=h(u,v)\) is

  \begin{equation*} \bv J_{\bv f}=\pdv{(x,y)}{(u,v)}= \begin{bmatrix} \DS\pdv{x}{u} \amp \DS\pdv{x}{v}\\[1 ex] \DS\pdv{y}{u} \amp \DS\pdv{y}{v} \end{bmatrix}. \end{equation*}

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• The Jacobian generalizes earlier notions of derivatives:

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  • \(\bb R \to \bb R\text{:}\) derivative

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  • \(\bb R \to \bb R^n\text{:}\) vector derivative

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  • \(\bb R^m \to \bb R\text{:}\) gradient (transpose)

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  • \(\bb R^m \to \bb R^n\text{:}\) full Jacobian

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• The determinant

  \begin{equation*} \det \bv J_{\bv f} = \pdv{x}{u}\pdv{y}{v} - \pdv{x}{v}\pdv{y}{u} \end{equation*}

  measures how small regions are scaled and whether orientation is preserved.

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• In two dimensions, this corresponds to the area of a transformed parallelogram.

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• Some transformations do more than scale—they preserve angles.

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• Example: \(f(z)=z^2\) corresponds to the transformation

  \begin{equation*} (x,y) \mapsto (x^2 - y^2, 2xy). \end{equation*}

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  Its Jacobian matrix is

  \begin{equation*} \begin{bmatrix} 2x \amp -2y\\ 2y \amp 2x \end{bmatrix}, \end{equation*}

  whose determinant is \(4(x^2 + y^2)\text{.}\)

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• The structure of this matrix shows that the transformation locally acts like a rotation combined with scaling.

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• In general, non-constant holomorphic functions produce transformations that preserve angles. These are called conformal mappings.

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