Maple does not have built-in commands for working with Fourier series. But you can download the app for this purpose here

Carl, the question was about the rotation of a loop of the curve . For example, a circle,  wholly located above the axis  Ox , rotates around the axis Ox. As a result, we get a torus. To get the volume of the torus, we must from body volume by rotation only the upper semicircle take away the body volume  by rotation only the lower semicircle ( r is the radius of the circle centered at the point  (0, R) ,  r<R ):

Sol:=solve(x^2+(y-R)^2=r^2, y);
f1:=unapply(Sol[1], x); f2:=unapply(Sol[2], x);
Int(Pi*f1(x)^2, x= -r..r) - Int(Pi*f2(x)^2, x= -r..r);
value(%) assuming r>0, R>0;

In  Student package:

Student[VectorCalculus][SurfaceInt](1, [x, y, z] = Surface(<r*cos(t), r*sin(t), r^2>, t = 0 .. 2*Pi, r = 0 .. 3));

By  int  command in polar coordinates:

F:=x^2+y^2:
int(r*eval(sqrt(1+diff(F, x)^2+diff(F, y)^2), {x=r*cos(t), y=r*sin(t)}), [r=0..3, t=0..2*Pi]);
expand(%);

These variants will suit you?

diff(f(x,y), x),  diff(f(x,y), y),  diff(f(x,y), x,y),  diff(f(x,y), x$2,y$3);

As you want:

`&PartialD;`*f(x,y)/cat(`&PartialD;`,x);
`&PartialD;`^2*f(x,y)/(cat(`&PartialD;`,y)*cat(`&PartialD;`,x));

Addition.

The simple procedure called  PartialD  solves your problem for an arbitrary number of variables:

restart;
PartialD:=proc(f, L)
local n, L1, L2;
uses ListTools;
L1:=Reverse([args[2..-1]]);
n:=nops(L1);
L2:=map(t->cat(`&PartialD;`,t),L1);
`&PartialD;`^n*f/convert(L2,`*`);
end proc:

Examples of use:

PartialD(f(x,y,z), x$2,y$3,z);

The arguments for the function, you don't need to write:

PartialD(H, x$2,y$3,z);

Edited.

error cannot be a name. A workaround:

restart;
`error`:=14/45*(D@@5)(y)(0)*h^5+7/10*(D@@6)(y)(0)*h^6;
remove(has, `error`, h^6);
                                      

For the plotting you must specify all the parameters. 

Example of the plotting:

restart;
schro := -(diff(psi(x), x, x))+(2*a*b*x^4+a^2*x^6+(b^2-a*(2*p+3))*x^2-(2*p+1)*b)*psi(x) = 0:
dsolve(schro);
assign(%):
plot(eval(psi(x), {a=1,b=1,p=1,_C1=2,_C2=1}), x=0..2);

multiply the latter matrix by 1/a^(3/4) :

A:=<a,b,c,d; e,f,g,h; i,j,k,l; m,n,o,p>;
B:=<a,b,c,d; 0,-e*b+a*f,-e*c+a*g,-e*d+a*h; 0,a*j-i*b,a*k-i*c,a*l-i*d; 0,-m*b+a*n,a*o-m*c,a*p-m*d>; 
D1:=LinearAlgebra[Determinant](A);
D2:=LinearAlgebra[Determinant](1/a^(3/4)*B);
D1-D2;

Download M.mw

plots[inequal]  command works for inequalities with 2 variables only. In your example 4 equalities with 3 variables are given. You can display them by plots[implicitplot3d]  command:

plots[implicitplot3d]([x-y-z=3, 2*x-y+2*z=5, x-y+z=1, 3*x-y-5*z=5], x=-7..7,y=-7..7,z=-7..7, color=[red,blue,green,yellow], style=surface, axes=normal,view=[-8.4..8.4,-8.4..8.4,-8.4..8.4]);

plots[inequal]  command works for inequalities with 2 variables. In your example 4 equalities with 3 variables are given. You can display them by   plots[implicitplot3d]  command:

plots[implicitplot3d]([x-y-z=3, 2*x-y+2*z=5, x-y+z=1, 3*x-y-5*z=5], x=-7..7,y=-7..7,z=-7..7, color=[red,blue,green,yellow], style=surface, axes=normal,view=[-8.4..8.4,-8.4..8.4,-8.4..8.4]);

Inequality  A>B  is equivalent to the inequality  A-B>0  and it will be true if

Optimization[Minimize](A-B, {constraints})[1] > 0

Examples:

A := 2*x^2+(y-2)^2-1: 
B := x^2:
Optimization[Minimize](A-B);  # On the whole plane (x,y) the inequality  A>B  does not hold 
Optimization[Minimize](A-B, {2*y <= x});  # At a given halfplane   2*y <= x  the inequality  A>B holds
is(%[1]>0);

Another way:

type(390/7, integer);

                                   false

I took  x0=0, y0=1 .

restart;
x0:=0: y0:=1:
plot([(-y0 - x0)*exp(-2*t) + (y0 + 2*x0)*exp(-t),  -2*(-y0 - x0)*exp(-2*t) - (y0 + 2*x0)*exp(-t), t=0..10], labels=[x,y]);

X:=(1/6)*sqrt(2)*sqrt(sqrt(13)+11):  Y:=-1/6+(1/6)*sqrt(13):
x:=y->piecewise(y<=1 and y>=Y, X/(Y-1)*(y-1), y<Y and y>=-Y, X, X/(-Y+1)*(y+1)):
T:=plots[textplot3d]([[1.35,0,0, x, align=below],[0,1.35,0, y, align=below],[0,0,1.15, z, align=right]], font=[times,roman,18]):
A:=plot3d([x(y)*cos(alpha), y, x(y)*sin(alpha)], y=-1..1, alpha=0..2*Pi, style=surface, color=khaki, view=[-1.4..1.4,-1.4..1.4,-1.2..1.2], axes=normal, orientation=[15,70,0]):
plots[display](A,T);

User lighting was used.

It can be done only for solving of a linear equation by Student[Basics][LinearSolveSteps] command. See help on this command.

Example of use:

Student[Basics] [LinearSolveSteps](2*x-1=1/3+x, x);

Do not use brackets to group terms in an expression. They are only used to create lists. 

plot(9.2-8.93*35.3/(11.17-8.93)*(exp(-8.93*t)-exp(-11.17*t))-8.3*exp(-11.17*t), t=0..1);