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Multivariable-Calculus · Divergence

Divergence of [sin(x)*cos(y), -cos(x)*sin(y), 0]

\nabla\cdot\left\langle \sin{\left(x \right)} \cos{\left(y \right)},\ - \sin{\left(y \right)} \cos{\left(x \right)},\ 0 \right\rangle

Step by step

  1. \mathbf{F} = \left\langle \sin{\left(x \right)} \cos{\left(y \right)},\ - \sin{\left(y \right)} \cos{\left(x \right)},\ 0 \right\rangle,\quad \nabla\cdot\mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}

    The divergence adds up how fast each component grows along its own axis.

  2. \frac{\partial P}{\partial x} = \frac{\partial }{\partial x}\left(\sin{\left(x \right)} \cos{\left(y \right)}\right) = \cos{\left(x \right)} \cos{\left(y \right)}

    Differentiate the x-component with respect to x, holding y and z constant.

  3. \frac{\partial Q}{\partial y} = \frac{\partial }{\partial y}\left(- \sin{\left(y \right)} \cos{\left(x \right)}\right) = - \cos{\left(x \right)} \cos{\left(y \right)}

    Differentiate the y-component with respect to y, holding x and z constant.

  4. \frac{\partial R}{\partial z} = \frac{\partial }{\partial z}\left(0\right) = 0

    Differentiate the z-component with respect to z, holding x and y constant.

  5. \nabla\cdot\mathbf{F} = \left(\cos{\left(x \right)} \cos{\left(y \right)}\right) + \left(- \cos{\left(x \right)} \cos{\left(y \right)}\right) + \left(0\right) = 0

    Add the three partial derivatives.

  6. \nabla\cdot\mathbf{F} = 0

    Zero divergence everywhere: the field has no sources or sinks. For a velocity field this is incompressible flow.

Reveal the answer
\nabla\cdot\mathbf{F} = 0