Mathquery

Green's Theorem:   Statement :-           If a domain E, regular with respect to both the axes , is bounded by a contour C , and f and g are two single - valued functions which along with their partial derivatives ∂f/∂y and ∂g/∂x  are continuous on E , then       ∫∫  (∂g/∂x - ∂f/∂y) dx dy = ∫ (f dx + g dy )        E                                         C  where the line integral is taken in the positive direction . Proof :-         Let us first consider a function f which , alongwith its partial derivative ∂f/∂y ,is continuous on a region E , regular with respect to y-axis . Let E be bounded by contour C , consisting of the curves y= φ(x) , y= ψ(x) , x = a , x = b , such that                        φ(x) ≤ ψ(x) , ∀ x ∈ [a,b] we have                    ∫∫ ∂f(x,y)/∂y dx dy                       E                                       b     ψ(x)                                   = ∫ dx ∫ ∂f(x,y)/∂y dy