Use  solve  instead of  fsolve:

restart;

solve({6.268 = a*c+0.710, 10.049 = a*(c+2.5*10^(-5)*(0.004/(0.025)))+0.710}, {a, c});

                                       {a = 945250., c =0 .5879925946e-5}

We can easily find maximum symbolically, if to convert the sum into the product and use that the maximum and minimum of  cos(a*t+b) , a<>0,  are the same as  cos(t) :

restart;

f:= cos(2*t/m) + cos(2*(t+5)/m):

simplify(applyrule(cos(alpha::anything)+cos(beta::anything)=2*cos((alpha+beta)/2)*cos((alpha-beta)/2), f));

max(eval(%, op(2,%)=-1), eval(%, op(2,%)=1));

restart;

eq1:=-sin(alpha0(t))*cos(beta0(t)) = -sin(alpha[1](t))*cos(beta[1](t));

eq2:=cos(alpha0(t))*cos(beta0(t)) = cos(alpha[1](t))*cos(beta[1](t));

simplify(eq1/eq2, trig);

convert(%, tan)*(-1);

This variant works immediately:

restart;

question1 := b(t)*(diff(c(t), t))*(diff(a(t), t))+a(t)*(diff(b(t), t)):

eval(question1, b(t)=0);

                                                0

See help on  Student[MultivariateCalculus][Jacobian]  command.

In your example, I have changed the variables so that they to be the names because

type(beta(t), name);

whattype(beta(t));

                                                 false

                                               function

Solution: 

eq1:=l1*cos(theta)+l2*sin(beta)-x=0:

eq2:=l1*sin(theta)-l2*cos(beta)=0:

Student[MultivariateCalculus][Jacobian](lhs~([eq1,eq2]), [beta,x,theta]);

T:=[alpha0(t)=alpha, beta0(t)=beta,psi[1](t)=psi, theta[1](t)=theta, gamma0(t)=gamma]:

A:=subs(T, eq_liaison):

add(applyrule([cos(u::anything)*cos(v::anything)-sin(u::anything)*sin(v::anything)=cos(u+v), cos(u::anything)*sin(v::anything)+sin(u::anything)*cos(v::anything)=sin(u+v), sin(u::anything)*sin(v::anything)-cos(u::anything)*cos(v::anything)=-cos(u+v), -sin(v::anything)*cos(u::anything)-sin(u::anything)*cos(v::anything)=-sin(u+v)], simplify(coeff(lhs(A),i), size))*i, i={xb[1],yb[1],zb[1],l2[1],l3[1]}):

B:=subs(map(t->rhs(t)=lhs(t),T), %);  # Final result

 Addition:

length(B);

                                          753

length(eq_liaison);

                                         1686

@Kimkimkim  Here is Konigsberg graph with 2 fictitious vertices  a  and  b , created as Carl has offered:

with(GraphTheory):

G:=Graph(Trail(1,2,3,4,1,3),Trail(1,a,2),Trail(1,b,4));

DrawGraph(G, style=planar);

@Markiyan Hirnyk   This example can be solved much shorter and with greater accuracy:

evalf(Int(exp(-a^3-b), [b=1-a..1+a, a=0..10]));

                     0.3707213163

You can use a caption instead of legend.

Example:

plot3d([(1/2)*x+(1/2)*y, 1+(1/2)*x+(1/2)*y], x = 0 .. 3, y = 0 .. 3, color = [yellow, blue], style = surface, axes = normal, orientation = [-30, 70], caption = "First plane is yellow, second plane is blue", captionfont = [Times, Roman, 14]);

Another variant - you can write the equations of the planes directly on the plot:

restart;

A := plot3d([(1/2)*x+(1/2)*y, 1+(1/2)*x+(1/2)*y], x = 0 .. 3, y = 0 .. 3, color = [green, blue], style = surface, axes = normal, orientation = [-30, 70]):

B := plots[textplot3d]([[3.3, 3.3, 2.5, z = (1/2)*x+(1/2)*y, color = green], [3.3, 3.3, 3.5, z = 1+(1/2)*x+(1/2)*y, color = blue]], font = [Times, Roman, 12]):

plots[display](A, B);

Edited.

Maple computes the integral right. But if you do not quite trust to Maple, you can do it step by step with a check these steps.

I dropped the constant factors as they do not affect the convergence of the integral. Also, we assume that  t> 0. The integrand in the interval  s=0 .. t  has only one singularity if  s=t. Therefore, we can consider the original integral as the limit of ordinary definite integrals on the range  s=0 .. p  when the variable  p  tends to  t .

A:=int(s^(5/2)/(t^(1/2)-s^(1/2)), s=0..p)  assuming t>0, p>0, p<t;

limit(A, p=t) assuming t>0;

Validation of calculation of the integral:

int(s^(5/2)/(t^(1/2)-s^(1/2)), s)  assuming t>0, s>0, s<t;

simplify(diff(%, s))  assuming t>0, s>0, s<t;

Without  solve - a line as an intersection of 2 planes:

L := [(x-2)*(1/3), (y-1)*(1/4), (z-3)*(1/3)]:

plots[intersectplot](L[1]-L[2], L[2]-L[3], x = -4 .. 4, y = -4 .. 4, z = -4 .. 4, linestyle = 1, thickness = 3, axes = normal, orientation = [60, 75], view = [-1 .. 3, -4 .. 2, -1 .. 4]);

Addition:  These planes  L[1]-L[2]=0  and  L[2]-L[3]=0  are the projecting planes on the coordinate planes x0y  and  y0z, respectively (the first one is perpendicular to x0y  and the second one is perpendicular to y0z). The equation  L[1]-L[2]=0  in the plane  x0y only, is the projection the original line into the plane  x0y  and  L[2]-L[3]=0  in the plane  y0z only, is the projection the original line into the plane y0z .

For  numeric  option, the parameters must be specified.

This version works for a small range:

restart;

dsys:={diff(u(t),t)=-(N1+M*cos(2*I*omega*t))*u(t)-1+v(t)*exp(-2*I*omega*t)+w(t)*exp(2*I*omega*t), diff(v(t),t)=-(N1+I*Delta-2*M*exp(2*I*omega*t))*v(t)-(N1+u(t))*exp(2*I*omega*t)-2*M, diff(w(t),t)=-(N1-I*Delta-2*M*exp(-2*I*omega*t))*w(t)-(N1+u(t))*exp(-2*I*omega*t)-2*M}:

N1,M,Delta,omega:=1,2,3,4:

res:=dsolve(dsys union {u(0)=-1,v(0)=0,w(0)=0},numeric,output=listprocedure, maxfun=500000);

plots[odeplot](res,[t,Re(w(t))],t=0..5,axes=boxed,color=black,linestyle=1,tickmarks=[3, 4],thickness=2);

 Addition:  Setting  maxfun=0  is dangerous if a singularity occurs, as in this case, the program can run forever. I think in your example of just such a situation.

LinearAlgebra[ColumnOperation]  command is not suitable to solve your problem.

Here is a short way without any packages:

A:=<1,-1,2,-1; 2,-2,3,-3; 1,1,1,0; 1,-1,4,3>;  # Construction of a matrix

b:=<-8,-20,-2,4>;  # Construction of a vector

A1:=<b | A[..,2..4]>;  # Replacement of the first column of a matrix

 Addition.  More 2 examples:

A2:=<A[..,1] | b | A[..,3..4]>;  # Replacement of the second column of a matrix

A4:=<A[..,1..3] | b >;  # Replacement of the last column of a matrix

solid of revolution for  (a) about x=-1:

restart;

with(plots): with(plottools):

ax := spacecurve([-1, t, 0], t = -1 .. 6, color = red, thickness = 2, linestyle = dash):  # Axis of rotation

s1 := animate(plot3d, [[(x+1)*cos(t)-1, x, -(x+1)*sin(t)], x = 0 .. 2, t = 0 .. alpha, style = surface, color = green], alpha = 0 .. 2*Pi, frames = 100):

s2 := animate(plot3d, [[3*cos(t)-1, y, -3*sin(t)], y = 2 .. 5, t = 0 .. alpha, style = surface, color = green], alpha = 0 .. 2*Pi, frames = 100):

s3 := animate(plot3d, [[(x+1)*cos(t)-1, x^2+1, -(x+1)*sin(t)], x = 0 .. 2, t = 0 .. alpha, style = surface, color = green], alpha = 0 .. 2*Pi, frames = 100):

s4 := animate(plot3d, [[cos(t)-1, y, -sin(t)], y = 0 .. 1, t = 0 .. alpha, style = surface, color = green], alpha = 0 .. 2*Pi, frames = 100):

display(ax, seq(s||i, i = 1 .. 4), scaling = constrained, lightmodel = light3, axes = normal, orientation = [60, 70], labels = [x, y, z]);

For  revolution  (b) about y=6, everything can be done similarly.

My procedure is not as elegant as Douglas B. Meade's one, but it does exactly what OP wanted. In addition, it works in older versions Maple in which there is no element-wise operation ~ :

Cat:=proc(S)

local S1;

uses ListTools;

S1:=[Categorize((x,y)->is(x[2]=y[2]), [op(S)])];

{map(t->`if`(nops(t)=1,op(t), [[seq(t[i,1],i=1..nops(t))],t[1,2]]), S1)[]};

end proc:

Example :

M10 := {[0,[1,4,2,1]],[1,[2,1,4,2]],[2,[3,6,4,3]],[3,[3,6,4,3]],[4,[4,2,1,4]],[5,[4,3,6,4]],[6,[4,3,6,4]],[7,[5,8,6,5]],[8,[5,8,6,5]],[9,[5,8,6,5]],[10,[6,4,3,6]]}:

Cat(M10);

{[0, [1, 4, 2, 1]], [1, [2, 1, 4, 2]], [4, [4, 2, 1, 4]], [10, [6, 4, 3, 6]], [[2, 3], [3, 6, 4, 3]], [[5, 6], [4, 3, 6, 4]], [[7, 8, 9], [5, 8, 6, 5]]}