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How can I introduce an arbitrary 4D metric in GRTENSOR, to calculate various components of corresponding Riemannian Tensor Components.

the question is as follow:

The partition does not always have to be equal intervals. Consider evaluating f(x)=x3 between 3 and 5, but splitting up the interval into a partition in which the end points of the subintervals are in a geometric progression. The common ratio r has to be chosen so that 3 is the first term and 5 is the last. Also the subintervals must be capable of getting smaller as n the number of subintervals increases. Check that the geometric series

a, ar, ar2, ar3,.....ari, .....arn =b

with r=  and suitable choices for a and b satisfies these criteria. Treating the difference between ari and ar(i-1) as the width of the subinterval and using the right hand endpoint of the subinterval, evaluate the Riemann sum to n terms for f(x)=x3. Find the limit as n tends to infinity to show that the partition does not affect the result.

here is what i have got so far, can anyone check if im doing it right? thanks

>a:=3:

>b:=a*r^10:

>r:=(5/3)^1/10:

>for i from 0 to 5 do a*r^i end do; -> a list of number appear in sequence ie:3, 3.157...,3.323...3.497...etc

>restart;

>a:=3:

>b:=a*r^100:

>r:=(5/3)^1/100:

>dxj:=a*r^i-a*r^i-1

>xj:=i*dxj+a

>f:=x->x^3

>evalf(sum(f(xj^*)dxj,i=1..100)) -> my value is sth like 162.4788870...

I tried to find the limit, but maple 16 freezed so i think i must have done sth seriously wrong?

<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>b</mi><mo>&coloneq;</mo><mi>a</mi><mo>&sdot;</mo><msup><mi>r</mi><mrow><mn>10</mn></mrow></msup><mo>&#x3b;</mo><mo>&nbsp;</mo></mrow></math>

Is it possible to change the caption of RiemannSum command in Student[Calculus1] package with animation output option, keeping approximate value and number of partitions outputs?  I tried to use typeset command, but I haven’t succeeded.

Best regards,

Emrah

 

Hey all,
 I have created a nested while loop to investigate the Riemann zeta function: 

for q to 10 do
 m := 10^(-q);
  for s from 2 to 10 do
   n := 1;
   b := 1;
    while b >= m do
     b := evalf(abs(Zeta(s)-euclid(s, n)));
     n := n+1
    end do;
  print(m, s, n-1)
 end do
end do 

it...

 

The attached worksheet is a wonderful introduction to the concept of obtaining the area under a curve.

You'll see how easy it is to learn how to find the limit of the sum of a series using Maple.

An interactive video tutorial that shows you how to do Riemann sums really fast is linked below:

(Ctrl+Click on the link to view the video)

Riemann Sums...

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