How to Calculate the Divergence of a Vector Field

The flow of magma underneath the earth's surface, the circulation of ionized particles within the core of the sun, the currents of water at the face of a dam -- you can describe all these with the mathematics of vector fields. A vector field consists of a set of magnitudes and directions, each tied to a specific location. For example, six miles beneath your feet, the magma may be moving 0.7 meters per second at 21 degrees east of north. Two hundred feet away, it may be flowing 0.4 meters per second at 13 degrees east of north. One of the most important parameters that describes the behavior of vector fields is the divergence.

Instructions

    • 1

      Separate each vector into its x, y, and z components.

      For example, the functional form of a vector field might be xy x-hat, x y-hat, (1-y)z^2 z-hat. The term "x-hat" refers to a unit vector in the direction of x; likewise for the other coordinates. So "xy x-hat" refers to a vector that has a magnitude of x times y in the direction of x.

    • 2

      Take the derivative of each component with respect to its direction. That is, take the derivative of the x-component with respect to x, and so on for the other components.

      The derivatives for the example function are (d/dx)(xy) = y, (d/dy)(x) = 0, (d/dz)(1-y)z^2 = 2z(1-y).

    • 3

      Add the derivatives together to get the divergence.

      For the example, the sum is y + 0 + 2z(1-y) = y + 2z(1-y).

    • 4

      Calculate the value of divergence for the values of interest in the field.

      In this example, for every value of x at y=0, the divergence is 2z, while for every value of x at y=1, the divergence is 1.

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