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 : Problems & Solutions :                                                                           This page having some problems on Double Integrals followed by solutions.                                                                                      1.      1.   Evaluate ∬   xy (x 2 +y 2 )dx dy over [0,a;0,a]. 2.       2 .  Evaluate   ∬ xy (x 2 +y 2 )dx dy over [0,a;0,b]. 3.        3 .  Evaluate  ∬  ye xy  over [0,a;0,b]. 4.      4 .  Ev...

 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...