@rit It seems to me that you are mixing up two different concepts:

1. Vector fields
   
   A vector field is something like [f(a,b,c), g(a,b,c), h(a,b,c)].  In Maple you apply fieldplot3d() to plot that vector field as a bunch of arrows, as you can see in the web page on fieldplot3d which you have quoted.  This is independent of any differential equations.
   
   Note that there is no "t" in here.
   
   
2. Differential equations
   
   Associated with that vector field there is a system of differential equations:
   
   da(t)/dt = f(a(t),b(t),c(t));
   db(t)/dt = g(a(t),b(t),c(t));
   dc(t)/dt = h(a(t),b(t),c(t));
   
   In Maple, you apply DEplot3d() to plot the solutions of that system of ODEs.  You will get a bunch of curves.

For instance, the vector field may be [a/r, b/r, c/r], where r=sqrt(a^2 + b^2 +c^2).  You may plot this vector field by applying fieldplot3d().

The associated system of differential equaitons is
          de1 := da(t)/dt = a(t)/r(t);
          de2 := db(t)/dt = b(t)/r(t);
          de3 := dc(t)/dt = c(t)/r(t);
You may plot the solutions curves with DEplot3d().

Now, what else do you want to do?

@Kitonum That's good.  Thanks!

@rit I suppose that the web page that you have found answers your question.  If you have other questions about it, then you should be more clear on exactly what it is that you are asking.

As to "in equations, there are a(t), b(t) and c(t), how to make it do not depend on t?"

The (t) that appear in those equations are just decorations.  What you have is exactly equivalent to:

r = sqrt(a^2+b^2+c^2);
fld := [a/r,b/r,c/r];

Not an answer but just a little advice to move you along.

1. Specify the domain of the functions w and phi.  In reading your PDF, I conjecture that you have meant 0 < x < 1 and -1/2 < z < 1/2.  Don't leave it to a guess.
2. Bad notation can slow you down.  Why don't you get rid of the bars above x, z, and w?  Probably they where introduced at some point while deriving your equations, but now that you have them, the bars serve no purpose and clutter the page and get in the way.
3. There is a better notation for specifying the boundary conditions.  In (42) and (43) replace the partial derivatives with index notation.  For instance, write w_xx for the second partial derivative of w with respect to x.
4. Drop the [...]_x=0,1 notation—it hides information and confuses.   Thus, write equation (41) as
   w(0,z) = 0,    -1/2 < z < 1/2
   w(1,z) = 0,    -1/2 < z < 1/2,
   which makes it clear that we have two equations, and that each of which should hold over the specified ranges of z. Fix the other boundary conditions similarly.
5. After you have applied step 4 to the boundary conditions (41) through (44), examine (42) and (44).  Wouldn't φ drop out of equation (42)?
   
   

@Carl Love Is there a variant of the construction ode||(1..3) that produces ode[1], ode[2], ode[3]?  I would normally do seq(ode[i], i=1..3) for that, but I wonder if there is a shorter way.

@rit The Wikipedia article that you have quoted pertains to vector fields.  Can you identify the vector field that you wish to examine?

Hint: As Preben Alsholm has pointed out in the earlier version of your question,  your system of differential equations is nonautonomous, therefore it does not define a (stationary) vector field, and consequently the concepts of source and sink are not applicable.

@mskalsi You have written u with a subscript x to denote the derivative of u with respect to x.  That's a common mathematical syntax, and therefore I understand what you mean, but that's not how Maple looks at things.  It expects diff(u(x,y),x) instead, which, among other things, makes it explicit that u is a function of x and y.  Similarly the mixed x and y derivative would be diff(u(x,y),x,y), etc.

Fix your worksheet and try again.

@tomleslie The derivation of the differentiation formula (A . B)' = A' . B + A . B' for matrices A an B is identical to that of the scalar version.  We have

[A(t+h) . B(t+h)  - A(t) . B(t)]/h
=
[A(t+h) - A(t)]/h . B(t)  +  A(t+h) . [B(t+h) - B(t)]/h.

We take the limit as h goes to zero and get

[A(t) . B(t)]' = A'(t) .B(t) + A(t) . B'(t).

@tomleslie I don't quite agree with your statement that "If you were to do this with pencil and paper, then you would proceed by comparing Matrix elements in your final expression. No other option is possible."

If I were do that with pencil and paper, I would do what the original poster has shown in the posted image, and would arrive at the equation boxed in red.  The matrix elements do not come into play.  We need not even know the matrix sizes. The boxed formua is true for any set of nxn matrices.  The point is, Maple is not capable of doing formulas like that.  Compare that with the scalar case which Maple handles with ease:

f := t-> g(t)*h(t);

diff(f(t),t);

@TomM I tried out your three execution groups (in Maple 2015.1) which altogether produce 4 plots.  The first is a thumbnail.  The last is full-sized.

Curiously, the two middle ones are variable-sized -- they grow and shrink as I resize the Maple window.  When I maximize the window, the two middle ones are actually larger than the fourth plot.  When I reduce the window size, the two middle one can shrink to be as small as a thumbnail.

This does not bother me, and until you brought this up, I was oblivious to the issue.

As to your last paragraph where you write "What I do not need is a lot of space being taken up by the thumbnail just to get this verification", I don't quite see what you are saying.  If you want verification, use semicolon.  If you don't, use colon.  But perhaps you have something else in mind.

As far as I can tell, what you are asking is not possible in Maple, unfortunately.  See this old discussion on the issue.

I think I recall seeing a behavior such as you have described, but I can no longer reproduce it.  Perhaps that is because I have upgraded from Maple 2015 to Maple 2015.1.

With Maple 2015.1's default configuration, p := plot(...); displays a thumbnail.  Subsequently, p; displays a normal-sized plot.

If you don't want to see the thumbnail, replace the semicolon with a colon, as in p := plot(...):

@max125 We know that the (-1)^x is +1 if x is even, and -1 if x is odd.  Now, what do you expect to get if x is neither odd nor even? Think of x = Pi.

@rit I don't understand the question at all.  Perhaps someone else who can understand it may respond.

I have no idea what it is that you are asking.  The expressions that you have shown are correct Maple syntax.  What do you want to know about them?