MVA b

1. F:O->R^n is called conservative if its line integral does not depend on path. If for any C1, C2 piecewise smooth paths in Omega with r,s:(a,b)->O with r(a)=s(a) and r(b)=s(b) we have that the line integrals of F.dr over C1 and F.ds over C2 are equal (12)
5. If F=nabla f and f is smooth then the line integral of F.dr over C does not depend on the path C but just on the end points. f is the potential of F. (9)
7. Let V be a solid in R^3 that is bounded by a piecewise smooth orientable S and n is the normal (outward). If F:V>R^3 is a smooth VF then sSSF.nds=vSSSdivFdxdydz (5)

2. If F is a vector field that is continuous on a connected set Omega subset of R^n then the following are quivalent: 1. There exists a scalar field such that F=nablaf, 2. F is conservative 3. The integral along a closed path is 0 (12, 3)
3. Integral of F.dr over a closed curve C = double integral of curlF.ndS over S (6, 7)
4. Let C be a piecewise smooth simple closed curve (SSCC) in the plane with param r[a,b]->R^2 that traveses C anti-clockwise and F is a smooth vector field defined in D vounded by C. Then work done = double integral over D dQ/dx-dP/dy dxdy (7, 7)
6. If r(a,b) is a smooth parametreisation of a path C and F:Omega -->r^n is a vector field such that F is defined and bounded on the image of r than we define the integral of F.dr over the path C as the integral between a and b of F(r(t)).r'(t)dt (4, 8)
8. double integral over omega of |r_a x r_b| dadB (4, 2, 1, 7)

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