I don't understand how the final animation in your worksheet is not what you want. To me, it shows both curves moving at the same time.

@ActiveUser I'm sorry, but I can't understand what you're trying to say about logic. 

You asked earlier what "degree" meant. The degree of a permutation is the length of the lists that it operates on. Since fixed points are omitted from the disjoint-cycle representation, the degree of a permutation in that form cannot be known with certainty, so it must be supplied for some commands. All that we can say for sure is that the degree is at least the largest number that appears.

All the permutations in a group must be the same degree, so, naturally, this is also called the degree of the group.

@chinthak I have very little knowledge of tensors, and no experience whatsoever of using them in Maple. If you post for me a worksheet that creates a small tensor, I may be able to figure out how to convert an Array to a tensor.

In other words, unless you're using an x that's already been defined rather unusally--as a function of two variables rather its usual meaning as an independent variable--then D[1,2](1/x) is pretty much garbage that is nonetheless (just barely) syntactically correct.

@MapleUser2017 Yes, I know.

@vv No, I meant "statement": Every procedure, arrow expression, or module is required to have a statement, possibly null. I didn't say that that could be any statement. In the case of an arrow expression, the statement must also be an expression, possibly null.

@Carl Love 

Kitonum has correctly pointed out that the tangent line does not exist for the specific example that you used. Nonetheless, what I said still holds: If you want to create multiple objects while overwriting the name, use the unevaluation quotes. 

@Magma Thank you. You've done a great job with the formatting. 

@Magma That technique is extremely inefficient, although it is unfortunately very commonly used, and even appears on some Maple help pages.

Whenever someone asks me to improve the efficiency of their Maple code (as you have done in another thread), that technique is the first thing that I look for.

@Stretto You've asked an excellent and subtle question. The answer is that local declarations and other stuff that may be in a procedure's header are not considered statements. Every procedure (or module or arrow expression) is required to have a statement, which could possibly be a null statement.

Also, it is a commonly held---yet erroneous---belief that statements need to be terminated with a semicolon or colon. This is even expressed on some Maple help pages. The truth is that (just like common English punctuation), those characters are used to separate statements.

@Stretto If S is your series, do

eval(S, sin= 0);

To understand how and why this syntax works: Any number, such as the 0 in this case, can be used as if it were a constant function.

Please supply your procedure in plaintext form. And it's about time that you learned to format and indent them properly. So instead of using some automatic plaintext conversion, please retype your procedure as indented plaintext. Why should I need to do that?

@Kitonum Yes, I was just trying to emphasize that the default definition of series convergence seems to be different between Maple and Mathematica, and that Maple's is the standard, Calculus II, definition.

I have no problem with there being an alternate definition named formal, or even multiple alternate definitions, as long as it's clear which definition is being used. 

You can't make a variable independent just by declaring it to be so. The equation that you show can be solved for r in terms of phi and Z. Doesn't that violate r's supposed independence?

Why can't you adapt the Answer given for your previous extremely similar Question to this new function? What trouble are you having with that? Show an attempt.

And why can't you stop posting poorly formatted output, like I asked before?