devotion & mathematics

    Bracket Function :                   The function I : R → Z defined by I(x) = n where n ≤ x < n+1 is called the bracket function or the step function or the integral part function .   Notation :                    Integral part of x in R is denoted by I(x) or [x]. Definition :                     If x 𝞊 R , x - [x] is called the Fractional part of x.    Note :         [x] ≤ x < [x] + 1  or x-1 < [x] ≤ x.       For every x 𝞊 R, x ≥ [x] ,  hence the fractional part of x is always non-negative       x 𝞊 Z ⇔ x=[x].   For example :         * For 14/3 = 4.6666... , [ 14/3] = 4.        * For 3/4, [3/4] = 0   because 0 < 3/4 < 1.        * For π = 22/7 , we...

                                                                                                    Contents :                                                                                  1. Vector function of a scalar variable          2. Limit of a vector function          3. Continuity of a vector function          4. Derivative of a vector function          5. Higher order derivatives          6. Propert...

              DOUBLE INTEGRALS               The line integrals are often used to find the lengths of the curves i.e if we talk about a circle the line integral is the circumference of the circle or semi circle or quarter circle etc. If we ask about the length of the curve on the circle which is not a complete circle or semi or quarte circle , we chose the part of the circumference of the circle then we need some procedure or  theory to find the length of the curve which are called the line integrals.          This process of line integration is applicable not only for the circles but also for any one or two or three dimensional regular  and irregular shapes .        So what about the surface area of circular cake in a rounded plate or cool drink bottle looks like a cylinder  or football like a sphere etc. In mathematics there are certain formulae for the ab...