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Multivariable-Calculus · Laplacian

Laplacian of sin(x)*sin(y)

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

Step by step

  1. f(x, y) = \sin{\left(x \right)} \sin{\left(y \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} = \sin{\left(y \right)} \cos{\left(x \right)},\quad \frac{\partial^2 f}{\partial x^2} = - \sin{\left(x \right)} \sin{\left(y \right)}

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

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

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

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

    Add the second partials.

Reveal the answer
\nabla^2 f = - 2 \sin{\left(x \right)} \sin{\left(y \right)}