devotion & mathematics

Compact Set   : A subset K of a metric space X is called compact if every open cover  has a finite sub cover. The compact sets has come properties .  Here we see their properties with proofs. Property 1. Closed Subsets Of Compact Sets are Compact . proof : Property 2 : The intersection of any collection of compact subsets of a metric space with finite intersection property is non-empty. Proof :   Property 3: Compact subsets of metric spaces are bounded. Proof :   property 4 : Compact subsets of metric spaces are closed. Proof :         People also see :  1. What is an open cover?  2. What is cantor intersection theorem ?  3. What is a metric space?  4. What is a closed set?  5. What is a Pseudo-metric ?  

Compact Sets In Metric Spaces :  Contents : * Open Cover of a metric space * Open Sub Cover of a metric space  * Compact Set  *  Examples                                                                                     Open Cover :                       Let K be a non-empty subset of a metric space X. A collection {G⍺/⍺ ꜪΔ} of                            open subsets of X is called an open   cover of K if K⊆⋃ G⍺.                       For example : Consider a metric space (R,d) where R is the set of all real                    ...

 Double Integrals : Integration over Rectangular Regions :                                                                           This page having some problems on Double Integrals followed by solutions.                                                                                                                         Back        1.   Evaluate ∬   xy (x 2 +y 2 )dx dy over [0,a;0,a]. SOLUTION :   2.       2 .  Evaluate   ...

 Repeated Integral   :                                  Let f(x) be a bounded function defined on the rectangle R=[a,b;c,d].            Suppose  ∫ f(x ,y) dy exists for each x  Ꜫ [a ,b] and y limits from c to d. If g(x) = ∫ f( x ,y )dy, y limits from c to d , then g(x) is a function on [a ,b] . If g(x) is also integrable on [a ,b] , then ∫g(x)dx =  ∫[  ∫ f(x , y)dy ] dx is called a repeated integral of f(x ,y) on R where x limits from a to b and y limits from c to d.                                                            Problems on Repeated Integrals : 1. Evaluate  ∫[  ∫ (x+y+1) dx ] dy and  ∫[ ∫(x+y+1) dy]dx , x limits from -1 to 1 and y limits are from -1 to 0. 2. Show tha...

              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...