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# 265L24 - LESSON 24 Stokes Theorem READ Section 18.2 NOTES...

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LESSON 24 Stokes’ Theorem READ: Section 18.2 NOTES: There are two operations related to the differential calculus of vector fields: the curl and the divergence . Like the gradient of a function of several real variables, these are vector field analogs of the derivative of a function of one variable. The names are rooted in their physical interpretations, which will be sketched a little later. For now, let’s just give the definition of the curl. In the next, and last, lesson, we will define the divergence. From now on, all vector fields will be in 3-space, so that -→ F ( -→ x ) = F 1 ( -→ x ) -→ ı + F 2 ( -→ x ) -→ + F 3 ( -→ x ) -→ k . Of course, by taking F 3 to be 0, what follows can be used for vector fields in 2-space as well. The curl of a vector field is another vector field. The definition is curl -→ F = ∂F 3 ∂y - ∂F 2 ∂z -→ ı + ∂F 1 ∂z - ∂F 3 ∂x -→ + ∂F 2 ∂x - ∂F 1 ∂y -→ k . The complicated looking formula for the curl is most easily remembered by thinking of the formal identity curl -→ F = ∇ × -→ F , where the formal cross product is computed using a determinant just as with vectors. See example 1, page 1001 of the text to see how to get the formula right. The curl shares some of the features of ordinary derivatives. For it is linear . That is, curl a -→ F + b -→ G = a curl -→ F + b curl -→ G.

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