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PFAFFIAN Differential Equations And It's Solutions

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PFAFFIAN Differential Equations And It's Solutions :-      The relation              n             Σ  Fᵢ(x₁,x₂,......xₙ) dxᵢ = 0            i=1   where Fᵢ's (i=1,2,....n) are functions of all or some of the n independent variables x₁,x₂,.....xₙ is called a Pfaffian differential equation . Here we shall study Pfaffian differential equation in three independent variables which is of the form              P dx + Q dy + R dz = 0    ...........(1)    where P, Q , R are functions of x , y and z . Setting -->                        -->               X = (P,Q,R)  and dr = (dx,dy,dz), equation (1) can be written in the vector notation as            ...
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Cauchy Riemann Equations For Analytic Function

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Cauchy - Riemann Equations :- Cauchy-Riemann Equation Necessary Conditions For A Function To Be Analytic : Cauchy-Riemann Equation Statement :- Cauchy-Riemann Equation     The necessary conditions for  w = f(z) = u(x,y) + i v(x,y) to be analytic (differentiable) at any point z = x + i y of its domain D is that the four patial derivatives ∂u/∂x , ∂u/∂y ,∂v/∂x ,∂v/∂y should exists and satisfy the C-R partial differential equations   .          ∂u/∂x = ∂v/∂y  and  ∂u/∂y = - ∂v/∂x Proof :- Cauchy-Riemann Equation       Let f(z) = u(x,y) + i v(x,y) be analytic at any point z of its domain , then               f'(z) = lim     f(z+δz) - f(z) / δz                         δx-->0         exists and is unique . i.e it i...
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Continuity And Differentiability In Mathematics | Mathquery

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          Welcome To Mathquery         In mathematics continuity and derivability or differentiability plays an important role . So let us discuss it . Definition Of Continuity Of A Function :-         Let f be a function defined on an interval [a,b] . We shall now consider the behaviour of f at points of [a,b] . Continuity At A Point :   Definition(Continuity At An Internal Point) :          A function f is said to be continuous at a point c , a<c<b , if                      lim f(x) = f(c)                    x-->c    In other words , the function is continuous at c , if for each ε>0 , there exists δ>0 such that               |f(x) - f(c) | < ε , when |x-c|<δ Here lim f...
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