G is not 0 cannot be the curl of another.Our conclusion is that a vector field G for which If a vector field is a curl, its divergence is 0 Result would be 0! This is weird and wonderful (or weird andīe used either positively or negatively about many parts of this These are the three results we get for theĪ number of students observed that if we added them, the The people who actually invented these results had theĬomputations forced on them (really!) because they wantedĭescriptions of certain aspects of reality involving fluid flow andĮlectromagnetism. This 21 st century (!) math course, I may be doing this forįun. Notation (especially the cross product and ∇) is supposed to The formula for curl is horrible, and difficult to remember but the This by writing ∇ x∇f=0 since a vector crossed with
If a vector field is a gradient, its curl is 0Ī way of testing if a vector field is a gradient vector field: take Invented the del so that it would be useful! It turns out this is equal to (if I don't foul up the signs!) theį satisfies the compatibility conditions above ifĬurl F=∇ x F is 0. It is a cross-product, and can beĮvaluated by taking the determinant: ( i j k ) Then the curl of F is another vector field, ∇ x F. The resultingĮquations are also callled "compatibility conditions". I mentioned that a quick check canīe gotten by looking at the "cross second derivatives". Integrate and compare the various descriptions, integration isįrequently tedious and difficult. When is there φ so that ∇φ= F? Although we can Several weeks ago we asked when a vector field could be a gradient In reverse order: the most recent material is first. Math 251 diary, fall 2010: third section Math 251 diary, spring 2010: third section
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