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revised What exactly is the difference between a derivative and a total derivative?
rewordings, included two more ways to say the same thing.
May
23
comment What exactly is the difference between a derivative and a total derivative?
If $x$ is secretly a function of $t$, then the notation $\frac{\text{d}}{\text{d}t} f(x,t)$ is called the total derivative and is an abbreviation for the (single-variable derivative) $g'(t)$ where $g(t)=f(x(t),t)$. In applying the chain rule to the last expression, you would need some way to denote "the derivative of $f$ with respect to its first argument" many people would write $\frac{\partial}{\partial x} f$ for this, but in many cases this is confusing.
May
23
comment Understanding why $a+b\sqrt {2}\neq \sqrt {3} $
@HagenvonEitzen what is the claim that you are referring to? I am saying that you could define the number we call sqrt(2) as the length of the diagonal of a unit square, right? My objection is to the claim that sqrt(2) is only defined algebraically as the root of something.
May
23
comment Understanding why $a+b\sqrt {2}\neq \sqrt {3} $
The length of a diagonal of a unit square is sqrt(2). The length of a diagonal of a unit cube is sqrt(3). That's how people discovered irrational numbers, as far as I can tell. It wasn't algebra.
May
23
comment What exactly is the difference between a derivative and a total derivative?
@HenningMakholm, I clarified and reorganized. I hope the point is clearer now! Since you are CS/PL person, I think you can help me make this even clearer. What I'm really getting at is some distinction between a function and an expression and named arguments versus positional arguments. People get confused because of a scoping issue. in the expression ∂f(x,y)/∂x, the x in the denominator is named argument of f. The x in the numerator is something else: a variable defined in the current scope.
May
23
revised What exactly is the difference between a derivative and a total derivative?
reorganization, clarification
May
22
comment What exactly is the difference between a derivative and a total derivative?
In my original answer, I have "the total derivative usually means..." Do you think I need to make this stand out typographically?
May
22
comment What exactly is the difference between a derivative and a total derivative?
I think one conclusion from the wall-o-text is that the concept of "total derivative" is due to sloppiness in math notation, and that we are better off without it. As such, I don't think there IS a clear definition. Do you see what I mean?
May
20
revised What exactly is the difference between a derivative and a total derivative?
typos and clarifications
May
20
revised What exactly is the difference between a derivative and a total derivative?
added 20 characters in body
May
4
comment connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian
I don't understand your second comment mentioning the constraint $(x-1)^2 \leq 0$. Is that in reference to an example in my question or is that your own example? I can sort of understand why the Lagrange multipliers would be wonky. If you tighten the left-hand-side of that constraint, then there is no $x$ that satisfies is
May
4
comment connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian
@Michael thanks for the comments. feel free to make an answer. The reason I say "fighting does not happen with linear objectives" (with linear penalty) is because I'm thinking of the big-$M$ method as a sort of penalty method. It's not exactly, because the penalty term is not squared. But you do choose an $M$ large enough to drive a variable to be zero (it can't go negative because of the non-negativity constraint in the standard form of a linear program).
Apr
29
revised connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian
edited tags
Apr
27
awarded  Promoter
Apr
27
revised connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian
added 13 characters in body
Apr
24
comment Trig and derivatives: If condition holds for derivative, does it hold for the original equation?
Glad to help. Please see math.stackexchange.com/help/accepted-answer
Apr
23
answered What exactly is the difference between a derivative and a total derivative?
Apr
23
accepted Singular Distribution
Apr
23
accepted logarithm of a matrix base a matrix — $\mathbf{A}^x = \mathbf{B}$
Apr
23
comment logarithm of a matrix base a matrix — $\mathbf{A}^x = \mathbf{B}$
This was long ago! Sorry. Yes, probably in my case, the right answer would have been to diagonalize both matrices (or perhaps use the SVD). If the the bases were the same, then there is a chance that the eigenvalues of $A$, after exponentiating by some integer, would equal the eigenvalues of $B$. It looks like what I was trying to do was find the period of the discrete-time linear time-invariant system. And so the approach would be to find $A=V^{-1}\Lambda V$ and the smallest integer $n$ such that $\Lambda^n = I$