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How To Multiply Fractions With Whole Numbers

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How To Multiply Fractions With Whole Numbers  :       Multiplication of two fractional numbers or rational numbers is  solved only in three steps i.e.       Before we discuss the method of how to multiply fractions we have to understand that what is a fractional number or a rational number ?       So let us go through definition, Definition Of Fraction or Rational Numbers :               A fraction or rational number is of the form p/q, where p and q are integers and q≠0.           Then here is the question arises that what is an integer?         To answer this question let us take examples,      The set of numbers like .............-3,-2,-1,0,1,2,3..........      are called as the set of integers.        Now come to our initial question of  h...
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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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Leibnitz's Rule Statement And It's Proof

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        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   ...
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Fubini's Theorem

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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             ...
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Cauchy's Integral Theorem | Mathquery

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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.     ...
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Green's Theorem | Mathquery

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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                  ...
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