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Multivariable-Calculus · Laplacian

Laplacian of log(x^2 y^2)

\nabla^2\left(\log{\left(x^{2} y^{2} \right)}\right)

Step by step

  1. f(x, y) = \log{\left(x^{2} y^{2} \right)},\quad \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}

    The Laplacian adds the second partial derivative in each direction.

  2. \frac{\partial f}{\partial x} = \frac{2}{x},\quad \frac{\partial^2 f}{\partial x^2} = - \frac{2}{x^{2}}

    Differentiate twice with respect to x, holding the other variables constant.

  3. \frac{\partial f}{\partial y} = \frac{2}{y},\quad \frac{\partial^2 f}{\partial y^2} = - \frac{2}{y^{2}}

    Differentiate twice with respect to y, holding the other variables constant.

  4. \nabla^2 f = \left(- \frac{2}{x^{2}}\right) + \left(- \frac{2}{y^{2}}\right) = - \frac{2}{y^{2}} - \frac{2}{x^{2}}

    Add the second partials.

Reveal the answer
\nabla^2 f = - \frac{2}{y^{2}} - \frac{2}{x^{2}}