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Continuity And Differentiability In Mathematics | Mathquery

- May 01, 2020

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(x) is called the limit of a function x-->c i.e. the function exists when x tends to c. (i) A function f is said to be continuous from the left at c if lim f(x) = f(c) x-->c-0 (ii) Als

Leibnitz's Rule Statement And It's Proof

- April 29, 2020

WELCOME TO MATHEMATICS I n this mathematics session I shall prove that , under suitable conditions, ' the derivative of the integral and the integral of the derivative are equal ' , and consequently , ' the two repeated integrals are equal for continuous functions '. Leibnitz's Rule In Mathematics: If f is defined and continuous on the rectangle R = [a,b;c,d] , and if (i) fₓ(x,y) exists and is continuous on the rectangle R , and d (ii) g(x) = ∫ f(x,y) dy , for x∈ [a,b] c then g is differentiable on [a,b] and d g'(x) = ∫ fₓ(x,y) dy c d d i.e., d/dx {∫ f(x,y) dy }=∫ ∂f(x,y)/∂x dy c c Proof Of Leibnitz's Rule In Mathematics : Since fₓ (∂f/∂x) exists on R ,

Fubini's Theorem

- April 27, 2020

Fubini's Theorem :- In the world of mathematics integration plays an important role . So we have discussed the advance form of integration which results the famous theorem Fubini's Theorem. Statement: If a double integral , I = ∫∫ f dx dy R exists over a rectangle R = [a,b;c,d] , and if d ∫ f dy also exists , for each fixed x in [a,b] , c b d then the iterated integral ∫ dx ∫ f dy exists a c and is equal to double integral I. Proof : Let ε be any positive number. Since the upper integral , Iᵘ is the infimum of the upper sums , there exists a partition P of R such that Σ Σ Mᵢⱼ ΔRᵢⱼ < Iᵘ + ε i j or Σ Σ Mᵢⱼ Δxᵢ Δyⱼ < Iᵘ + ε ...........(1)

Cauchy's Integral Theorem | Mathquery

- April 25, 2020

Cauchy's Integral Theorem: Statement : The theorem is usually formulated for closed paths as follows : Let U be an open subset of C which is simply connected . Let f: U-->C be a holomorphic function , and let γ be a rectifiable path in U whose start point is equal to its end point . Then ∮ f(z) dz = 0 γ Proof : Let us assume that the partial derivatives of a holomorphic function are continuous , the Cauchy Integral Theorem can be proved as direct sequence of Green's Theorem and the fact that the real and imaginary parts of f = u + i v must satisfy the Cauchy - Riemann equations in the region bounded by γ , and moreover in the open neighbourhood U of this region. Cauchy provided this proof , but it was later proved by Goursat without requiring techniques from vector calculus , or the continuity of partial derivatives . We can brea

Green's Theorem | Mathquery

- April 24, 2020

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

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