I have just begun to study Green's functions and made some initial progress on a problem, but now need Maple to make further progress.  Apologies, I have written up the equations in LaTeX form rather than Maple, as my Maple has gotten very rusty.

$\frac{d^4y}{dx^4}=f(x)

y(0)=y'(0)=0, y(1)=y'(1)=0$

I showed that the Green function $G(x,u)$ for this equation satisfies a condition

$\lim_{\epsilon\to0}\bigg[\frac{\partial^3G}{\partial x^3}\bigg]_{x=u-\epsilon}^{u+\epsilon}=1$

and showed that there is continuity of the Green's function and its first and second partial derivatives with respect to $x$ at $x=u$.  The next step is to show that this function has a piecewise definition such that

\[G=\frac{1}{6}x^2(1-u)^2(3u - 2ux -x\] for x between the range 0 and u and such that

\[G(x,u)=\frac{1}{6}u^2(1-x^2)(3x - 2xu - u) for x between u and 1

I am not entirely sure how to do this with pen and paper, so I have reason to believe that it could be a done a lot more easily with Maple, if someone could give some pointers that would be much appreciated.

I am attempting to write a series representation of a general integral of a function from a to b as follows:

int(f(x), x = a..b)= h*sum((c_k)*f(a+kh))+O(h^p),k=1..N;

where h:=(b-a)/(N+1), p(N) is greater than or equal to N + 1 and c_k are coefficients.  I then need to write procedures with Maple to evalue c_k from 1,..,N and also to evaluate P(N) for any N.  If I take the case for N = 3 and N = 6 I have to use those procedures to prove that:

int(f(x), x = a..b)=(4h/3)*(2*f_1 - f_2 +2*f_3) + O(h^5) = (7*h/1440)*(611*(f_1 + f_6) - 453*(f_2 + f_5) + 562*(f_3 + f_4)) + O(h^7) 

where f_k = f(a + kh).  I am really at a loss as to how to write this procedure, although I may have used something similar before:

P:=proc(p) add((1/k^(1/10))*(sin(1/k)-1/k), k=1..10^p) end proc;
seq( evalhf(P(p)), p = 1 .. 5 );
 

FLRW_Metric.mw

I have been tasked with calculating all the non-vanishing Christoffel symbols (first kind) of a metric and have done these long-hand using the Lagrangian method and shown my working. However, for peace of mind I would like to run the metric through Maple and double-check that it returns the same answers (going back through my calculations if I have missed anything). I have attached the code I have written at the bottom.

I have no trouble defining the metric and the manifold but I receive an error message when I try to compute the Christoffel symbols 'improper op or subscript selector'. Could someone point out where I have made a mistake. The metric is the FLRW metric if that helps.

with(DifferentialGeometry):with(Tensor);

g1:=evalDG(-(dt)^2 +a(t)^2*((dx)^2+(dy)^2+(dz)^2)/(1+(k/4)*(x^(2)+y^(2)+z^2))^2 );

C1:=Christoffel(g1, "FirstKind");

I am studying the sum below and am trying to write a procedure with Maple which I can use to efficiently evaluate the sum for large values of n.

sum((1/k^0.1)*sin(1/k), k=1..n)

I have read the section on procedures in my textbook for Maple but it only has very simple examples with nothing relevant to this situation and I am struggling to apply it here: can anyone assist?

I am considering a function y(x) which is defined implicitly such that:

y(x):= x-> ln((1+x)*y) + exp(x^(2)*y^(2))

I am attempting to show that the Taylor series about x = 0 is:

y(x) = 1 - x^2 - (1/3)x^3 + (55/24)x^4 + (4/5)x^5 - (439/80)x^6 + O(x^7)

This could be done by using Maple to take derivatives up to sixth order and then substituting into the formula for the Taylor series.  My initial idea for the Taylor series was to use the identity 

ln((1+x)*y) + exp(x^(2)*y^(2)) = x + cos(x)

but obviously the RHS does not give the required form for the Taylor series about x = 0 (although I can use it to show that y(0) = 1).  Would it be possible to use Maple to take six derivatives of the left-hand side instead and then use that to create the Taylor series as far as sixth order?