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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}{x^{2} + y^{2}},\quad \frac{\partial^2 f}{\partial x^2} = - \frac{4 x^{2}}{\left(x^{2} + y^{2}\right)^{2}} + \frac{2}{x^{2} + y^{2}}

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

  3. \frac{\partial f}{\partial y} = \frac{2 y}{x^{2} + y^{2}},\quad \frac{\partial^2 f}{\partial y^2} = - \frac{4 y^{2}}{\left(x^{2} + y^{2}\right)^{2}} + \frac{2}{x^{2} + y^{2}}

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

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

    Add the second partials.

  5. \nabla^2 f = 0

    Simplify.

  6. \nabla^2 f = 0

    The Laplacian is zero, so f is harmonic: it solves Laplace's equation.

Reveal the answer
\nabla^2 f = 0