devotion & mathematics

  LINE INTEGRALS - PROBLEMS &SOLUTIONS :       We have discussed about the definitions and development of line integrals .  Here we going to solve some problems regarding line integrals.  PROBLEM 1 :  .   Evaluate  ∫ dx/(x+y),  Where C  is the curve x=at 2 ,y=2at, 0 ≤ t ≤ 2. SOLUTION :                          PROBLEM 2 :   Show that ∫ xy dx  =4/5, where C is the arc of the parabola y 2 =x from (1,-1) to   (1,1). SOLUTION :                         3.   PROBLEM 3 :  Show that ∫  [ (x-y) 3 dx+(x-y) 3 dy]=3 πa 4 , where C Is the circle x 2 +y 2 =a 2  in       the counter clockwise sense.         SOLUTION :                                ...

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...