6.5 Divergence and curl  (Page 5/9)

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 $m_1$ at the origin and an object with mass $m_2.$ Recall that the gravitational force that object 1 exerts on object 2 is given by field

Using divergence and curl

Now that we understand the basic concepts of divergence and curl, we can discuss their properties and establish relationships between them and conservative vector fields.

If F is a vector field in ${ℝ}^3,$ then the curl of F is also a vector field in ${ℝ}^3.$ Therefore, we can take the divergence of a curl. The next theorem says that the result is always zero. This result is useful because it gives us a way to show that some vector fields are not the curl of any other field. To give this result a physical interpretation, recall that divergence of a velocity field v at point P measures the tendency of the corresponding fluid to flow out of P . Since $\text{div}\text{curl}\left(\text{v}\right)=0,$ the net rate of flow in vector field curl( v ) at any point is zero. Taking the curl of vector field F eliminates whatever divergence was present in F .

Proof

By the definitions of divergence and curl, and by Clairaut’s theorem,

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With the next two theorems, we show that if F is a conservative vector field then its curl is zero, and if the domain of F is simply connected then the converse is also true. This gives us another way to test whether a vector field is conservative.

Proof

Since conservative vector fields satisfy the cross-partials property, all the cross-partials of F are equal. Therefore,