E33.2 A friend tells you that if S is a closed surface (that is, a surface without a boundary curve), then by Stokes' theorem we ought to have [/, s curl F-dS = 0 for any appropriate F' (since there is no boundary curve C). Is this true? Can you justify it using the Divergence theorem? Let's start with the curl. Given the vector field F = P 7 + Q 3 + RF the curl is defined to be, curlF = (R, Q,) i + (P, R,) j + (Q. B,) & There is another (potentially) easier definition of the curl of a vector field. To use it we will first need to define the V operator. This is defined to be, We use this as if it's a function in the following manner. _9F ;005 0y v f 6'.7:2+0y] 0z So, whatever function is listed after the V is substituted into the partial derivatives. Note as well that when we look at it in this light we simply get the gradient vector. Using the V we can define the curl as the following cross product, curlzVXz Let's now talk about the second new concept in this section. Given the vector field f =Pi + Q} + Rk' the divergence is defined to be, 0Q OR L There is also a definition of the divergence in terms of the V operator. The divergence can be defined in terms of the following dot product. The final topic in this section is to give two vector forms of Green's Theorem. The first form uses the curl of the vector field and is, where E is the standard unit vector in the positive z direction. The second form uses the divergence. In this case we also need the outward unit normal to the curve C. If the curve is parameterized by r)=z(t)i+y(t)J then the outward unit normal is given by, y) - Z(t) - Fol Fol ',i': Divergence Theorem Let E be a simple solid region and S is the boundary surface of F with positive orientation. Let F' be a vector field whose components have continuous first order partial derivatives. Then, [f.d:f}!/dwdv