![]() ![]() In particular, if the amount of fluid flowing into P is the same as the amount flowing out, then the divergence at P is zero. If F represents the velocity of a fluid, then the divergence of F at P measures the net rate of change with respect to time of the amount of fluid flowing away from P (the tendency of the fluid to flow “out of” P). ![]() Locally, the divergence of a vector field F in ℝ 2 ℝ 2 or ℝ 3 ℝ 3 at a particular point P is a measure of the “outflowing-ness” of the vector field at P. Divergenceĭivergence is an operation on a vector field that tells us how the field behaves toward or away from a point. ![]() In addition to defining curl and divergence, we look at some physical interpretations of them, and show their relationship to conservative and source-free vector fields. For example, under certain conditions, a vector field is conservative if and only if its curl is zero. We can also apply curl and divergence to other concepts we already explored. In addition, curl and divergence appear in mathematical descriptions of fluid mechanics, electromagnetism, and elasticity theory, which are important concepts in physics and engineering. They are important to the field of calculus for several reasons, including the use of curl and divergence to develop some higher-dimensional versions of the Fundamental Theorem of Calculus. In this section, we examine two important operations on a vector field: divergence and curl. 6.5.3 Use the properties of curl and divergence to determine whether a vector field is conservative.6.5.2 Determine curl from the formula for a given vector field.6.5.1 Determine divergence from the formula for a given vector field. ![]()
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