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Multivariable-Calculus · Laplacian

Laplacian of 1/sqrt(x^2+y^2+z^2)

\nabla^2\left(\frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}}\right)

Step by step

  1. f(x, y, z) = \frac{1}{\sqrt{x^{2} + y^{2} + z^{2}}},\quad \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}

    The Laplacian adds the second partial derivative in each direction.

  2. \frac{\partial f}{\partial x} = - \frac{x}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}},\quad \frac{\partial^2 f}{\partial x^2} = \frac{3 x^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} - \frac{1}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}

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

  3. \frac{\partial f}{\partial y} = - \frac{y}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}},\quad \frac{\partial^2 f}{\partial y^2} = \frac{3 y^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} - \frac{1}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}

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

  4. \frac{\partial f}{\partial z} = - \frac{z}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}},\quad \frac{\partial^2 f}{\partial z^2} = \frac{3 z^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} - \frac{1}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}

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

  5. \nabla^2 f = \left(\frac{3 x^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} - \frac{1}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\right) + \left(\frac{3 y^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} - \frac{1}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\right) + \left(\frac{3 z^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} - \frac{1}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}\right) = \frac{3 x^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} + \frac{3 y^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} + \frac{3 z^{2}}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{5}{2}}} - \frac{3}{\left(x^{2} + y^{2} + z^{2}\right)^{\frac{3}{2}}}

    Add the second partials.

  6. \nabla^2 f = 0

    Simplify.

  7. \nabla^2 f = 0

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

Reveal the answer
\nabla^2 f = 0