Divergence and curl of a vector field.

18. Div grad curl and all that Theorem 18.1. Let A ⊂ Rn be open and let f : A −→ R be a differentiiabl function. If �r : I −→ A is a flow line for �f : A −→ Rn, then the function f ◦ �r : I −→ R is increasing. Proof. By the chain rule, d(f �r) dt ◦ (t)= �f(�r(t)) · �r�(t) = �r�(t) �r�(t) ≥ 0. �· Corollary 18.2. A closed parametrised curve is never the flow line of a conservative vector field. Once again, note that (18.2) is mainly a negative result: Example 18.3. F�: R2−{(0, 0)} −→ R2 given by F�(x,y)=(−x2 + yy,x2 + xy),22 is not a conservative vector field as it has flow lines which are circles. Definition 18.4. The del operator is the formal symbol ∂∂∂ ˆ� =ˆı + jˆ+ k. ∂x ∂y ∂z Note that one can formally define the gradient of a function grad f : R3 −→ R3 , by the formal rule ∂f ∂f ∂f ˆgrad f = �f = ∂x ˆı + ∂y jˆ+ ∂z k. Using the operator del we can define two other operations, this time on vector fields: Definition 18.5. Let A ⊂ R3 be an open subset and let F�: A −→ R3 be a vector field. The divergence of F�is the scalar function, div F�: A −→ R, which is defined by the rule ∂f ∂f ∂f div F� (x, y, z) = � · F� (x, y, z) = ∂x + ∂y + ∂z . 1 ������� � � � � � ������� �������������The curl of F�is the vector field curl F�: A −→ R3 , which is defined by the rule curl F�(x,x,z)= �× �F (x,y,z)������ ∂F∂F∂F∂F∂F∂F323121 ˆˆıjˆk. +−−−−= ∂y ∂z ∂x ∂z ∂x ∂y Notethatthedeloperatormakessenseforany ,notjust 3.nn = Sowecandefinethegradientandthedivergenceinalldimensions. Howevercurlonlymakessensewhen =3. n�3RDefinition18.6. Thevectorfield FA iscalled rotation : −→ free if the curl is zero, curl F�= �0, and it is called incompressible if the divergence is zero, div F�=0. Proposition 18.7. Let f be a scalar field and F�a vector field. (1) If f is C2, then curl(grad f)= �0. Every conservative vector field is rotation free. (2) If F�is C2, then div(curl F�)=0. The curl of a vector field is incompressible. Proof. We compute; ˆıjˆkˆ∂∂∂=∂x ∂y ∂z F1 F2 F3 curl(grad f)= �× (�f) ˆıjˆkˆ∂∂∂ =∂x ∂y ∂z ∂f ∂f ∂f ∂x ∂y ∂z ∂2f∂2f ∂2f∂2f ∂2f∂2f kˆˆjˆ+ı − ∂x∂z − ∂z∂x ∂y∂z −∂z∂y∂x∂y −∂y∂x== �0. 2 ������ ������������������This gives (1). div(curl F�)= �· (�× f) = �·ˆıjˆkˆ∂∂∂ ∂x ∂y ∂z F1 F2 F3 =∂ ∂ ∂ ∂x ∂y ∂z ∂ ∂ ∂ ∂x ∂y ∂z F1 F2 F3 ∂2F3 ∂2F2 ∂2F3 ∂2F1 ∂2F2 ∂2F1 = ∂x∂y − ∂x∂z − ∂y∂x + ∂y∂z + ∂z∂x − ∂z∂y =0. This is (2). � Example 18.8. The gravitational field �cx cy cz ˆF (x,y,z)= ˆı+ jˆ+ k, (x2 + y2 + z2)3/2 (x2 + y2 + z2)3/2 (x2 + y2 + z2)3/2 is a gradient vector field, so that the gravitational field is rotation free. In fact if c f(x,y,z)= −(x2 + y2 + z2)1/2 , then F�= grad f, so that curl F�= curl(grad f)= �0. Example 18.9. A magnetic field B�is always the curl of something, B�= curl �A, where A�is a vector field. So div( B�) = div(curl A�)=0. Therefore a magnetic field is always incompressible. There is one other way to combine two del operators: Definition 18.10. The Laplace operator take a scalar field f : A −→ R and outputs another scalar field �2f : A −→ R. It is defined by the rule ∂2f∂2f∂2f �2f = div(grad f)= ∂x + ∂y + ∂z . 3 A solution of the differential equation �2f =0, is called a harmonic function. Example 18.11. The function c f(x,y,z)= −(x2 + y2 + z2)1/2 , is harmonic. 4 MIT OpenCourseWarehttp://ocw.mit.edu 18.022 Calculus of Several Variables Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.