@Bohdan The problem is that you're using 2D input.

@a_simsim Your dsolve command should be

sol:= dsolve({odesys[], ics[]}, numeric, known= Q(t));

@Bohdan I only suggested Physics:-Library:-Add because it may be able to "pretty print" the summations with the inequalities at the bottom. Computing the formulas is straightforward with regular add, and shown below. If you do want to do that pretty printing, you should ask a separate Question. Here is your formula A:

MyA:= (
    i::posint, r::posint, j::posint, s::posint, F::posint,
    J::posint, S::posint,
    gamma::{name, table}, beta::{name, table}, pi::{name, table},
    alpha::{name, table}, epsilon::{name, table}
)->
local
    Jj:= {$1..J} minus {j},
    Omega:= zz->
        add(
            alpha[k,s]*pi[j,k,s,F]*
            `if`(nargs=0, add(pi[z,k,s,F], z= Jj), pi[zz,k,s,F])*
            add(gamma[k,u]*beta[k,u], u= 1..S)
            +
            add(
                gamma[k,s,u]*beta[k,u]*pi[j,k,s,u]*
                `if`(nargs=0, add(pi[z,k,s,u], z= Jj), pi[zz,k,s,u]),
                u= 1..S
            ),
            k= 1..J
       )
;
    epsilon[s]*gamma[i,r]/beta[j,s]*(
        Omega()*(pi[i,j,r,s]*gamma[j,r,s] + `if`(r=s, `if`(i=j, 1, -1), 0)) +
        add (pi[i,z,r,s]*gamma[z,r,s]*Omega(z), z= Jj)
    )
;     

Regarding the derivative of the formula: Do you mean when J, S, and the indices of the differentiation variable have specific integer values? If so, then plain diff will work.

Does that worksheet work for you? And, if so, what was the problem before?

@ecterrab I agree about the potential confusion caused by Maple's type algebraic, and so it would likely be best to avoid the word. (Although, I wish that that type name were changed! Type algebraic is much more general than the mathematical concepts algebraic numbers, algebraic functions, or algebraic equations (as in "system of differential-algebraic equations (DAEs)). That's never going to happen though.)

Regarding not versus non-: Not is an adverb, and in this case it modifies the verb can include. So, as it stands, the help-page phrase is equivalent to "can include ODEs and cannot include differential equations", and this is how nm understood it. To negate an adjective such as differential, use the prefix non-, which may also be attached without a hyphen.

I deleted a duplicate of this Question, whose only difference seemed to be different names of the variables. 

@janhardo A restart command should always be placed in its own execution group (its own prompt). While it usually doesn't cause problems if it's mixed with other commands, when it does, those problems are usually GUI related and difficult to predict. This is because the GUI and kernel run asynchronously. This is my first suspicion for the restart problem that you describe, but it's only a suspicion.

@nm The help-page phrase that you asked about, "...can include ODEs and not differential equations", should be "...may include ODEs or non-differential equations" or "...may include ODEs or algebraic equations". Does that clear up the confusion for you? And, if so, which of the two alternative rephrasings is clearest?

@9009134 Here it is:
 

Download GradPlot.mw

@janhardo The mathematical meaning of path integral is not completely standardized, especially in other languages. I think that to be completely clear, one must specify either scalar line integral or vector line integral.

And neither of these should be called a surface integral, which is an integral whose region of integration is a surface. The result of a scalar line integral can be interpreted as a difference of surface areas above and below the xy-plane, but that is not reflected in the integral's name.

The Wikipedia article "Line integral" has an excellent animation to illustrate this surface-area interpretation. 

@9009134 Just adding code is not enough. You must follow the steps that I specified, in the order that I specified:

1. Remove some lines.

2. Add some lines.

3. Re-execute the entire worksheet with the !!! button.

The error that you're seeing is due to aspects of the Physics[Vectors] subpackage still being active. The issue is not due to a bug in that package; rather, the package is simply inappropriate for this purpose.

What you want to do is possible.

But it's not possible that the code you show above, by itself, could produce the output that you show. Please provide the complete code that produced that output.

@David Sycamore Add the option

axis[2]= [mode= log]

inside the parentheses of the plotting command. I'm not sure if this will work in your version of Maple. Let me know.

Unfortunately, int(exp(a*t^b), t) ​​​​​​for explicit real constants a and b returns in vastly different forms depending on a and b. Sometimes it returns WhittakerM functions, sometimes it returns incomplete GAMMA functions (which are much simpler), and sometimes it returns unevaluted (such as for a=1, b= Pi). Can it be taught to always return the GAMMAs?

@Carl Love I want to emphasize that I am not claiming that the vector-field plot above has any relevance to the physical problem described. It simply illustrates the steps that one must take to plot the gradient of a 2-variable function.