Science Quiz / Theorems In Calculus

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Statement DescriptionTheorem
Let C be a simple closed curve and f differentiable from R^2 to R^2, then Int(f,C,s) = Int^2(curl(f), I(C),S) where I(C) is the interior of C
Let H be a surface in R^3 with a boundry d(H), a simple closed curve. Let f be differentiable form R^3 to R^3, then Int(f,d(H),s) = Int^2(curl(f),H,S)
Let H be a surface in R^3 which encloses a volume, I(H). Then for f differentiable from R^3 to R^3, Int^2(f,H,S) = Int^3(div(f),I(H),V)
Let f:C->C be holomorphic, then for all paths p, Int(f(z),p,z) = 0.
Let F:C->C be holomorphic in D and let p be a path in D connecting z0 and z1. Then Int(F'(z),p,z) = F(z1)-F(z0)
Let f be continuous on a simply connected domain. Then the following are equivalent; i) If a path p is closed then Int(f(z),p,z) = 0, ii) there exists F(z), holomorphic, such that
Let f be holomorphic in A and let p be a closed curve in A. Suppose we can continuously deform p to q in A. Then Int(f,p,s) = Int(f,q,s)
Let C be a simple closed curve. Let f be holomorphic over C and its interior I(C) except for n+1 singularities z(0),z(1),...,z(n). Then, Int(f(z),C,z) = 2*pi*sqrt(-1)*SUM(Res(f,z(i
Let x be an isolated singularity of f. Then there exists E>0 such that f is holomorphic in the E-Neighbourhood of x when f has the representation, f(z) = SUM(a(k)*(z-x)^k, k=...-1,
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