Recall that the gravitational force that object 1 exerts on object 2 is given by fieldįield v ( x, y ) = 〈 − y x 2 + y 2, x x 2 + y 2 〉 v ( x, y ) = 〈 − y x 2 + y 2, x x 2 + y 2 〉 models the flow of a fluid. We can now use what we have learned about curl to show that gravitational fields have no “spin.” Suppose there is an object at the origin with mass m 1 m 1 at the origin and an object with mass m 2. Using curl, we can see the circulation form of Green’s theorem is a higher-dimensional analog of the Fundamental Theorem of Calculus. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function f f on line segment can be translated into a statement about f f on the boundary of. If we think of curl as a derivative of sorts, then Green’s theorem says that the “derivative” of F on a region can be translated into a line integral of F along the boundary of the region. Therefore, the circulation form of Green’s theorem can be written in terms of the curl. Where C is a simple closed curve and D is the region enclosed by C. Similarly, div v ( P ) < 0 div v ( P ) < 0 implies the more fluid is flowing in to P than is flowing out, and div v ( P ) = 0 div v ( P ) = 0 implies the same amount of fluid is flowing in as flowing out.
![divergence in curved space divergence in curved space](https://miro.medium.com/max/1104/0*axBuH2vxM9CQYhRe.png)
Since the divergence of v at point P measures the “outflowing-ness” of the fluid at P, div v ( P ) > 0 div v ( P ) > 0 implies that more fluid is flowing out of P than flowing in. Let v be a vector field modeling the velocity of a fluid. We can use all of what we have learned in the application of divergence. Using divergence, we can see that Green’s theorem is a higher-dimensional analog of the Fundamental Theorem of Calculus. This is analogous to the Fundamental Theorem of Calculus, in which the derivative of a function f f on a line segment can be translated into a statement about f f on the boundary of. If we think of divergence as a derivative of sorts, then Green’s theorem says the “derivative” of F on a region can be translated into a line integral of F along the boundary of the region. Therefore, Green’s theorem can be written in terms of divergence. 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.
![divergence in curved space divergence in curved space](https://s1.cdn.autoevolution.com/images/news/gallery/bmw-announces-idrive-8-with-curved-displays-upgraded-assistant-modern-ui_38.jpg)
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.