3
votes
1answer
105 views

What does “calculus” mean?

"calculus" and "formal system" From http://en.wikipedia.org/wiki/Propositional_calculus#Terminology a calculus is a formal system that consists of a set of syntactic expressions ...
2
votes
0answers
64 views

Please identify this equation: $\nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A$

Is this equation $$ \nabla^2 \mathbf F -k^2 \mathbf F = \mathbf A $$ somehow named? F and A are vector fields. I guess inhomogeneous sign reversed Helmholtz equation isn't appropriate ...
6
votes
1answer
56 views

What is the difference between a calculus and an algebra? [duplicate]

You can have a lambda calculus, the calculus of the real numbers or a logical calculus but on the other hand you could also have an algebra of sets, a Lie algebra, or a linear algebra. Is there any ...
0
votes
2answers
91 views

Why do some sources call calculus, “the calculus”?

No need to cite specific sources since I think it's a fairly common thing to see. What's up with that? Thank you Edit: I've seen it in several places. Here's where I'm currently looking at it at: ...
0
votes
1answer
16 views

Difference between $C_0^{\infty}(U)$ with support in $A$ and $C_0^{\infty}(A)$

Let $A \subseteq U$ be open sets of $\mathbb R^n$. Is it true that $$ \lbrace f \in C_0^{\infty}(A) \rbrace = \lbrace f \in C_0^{\infty}(U) : \text{support of } f \subseteq A \rbrace \quad ? $$ I ...
0
votes
1answer
18 views

Critical points of a function of absolute value

Say I have the function $f(x) = |x|$ I believe that $x = 0$ is a critical point, although not I'm not positive. As the function is decreasing and increasing each side of $x = 0$ does that alone make ...
1
vote
2answers
40 views

How to find if the sum of periodic function is periodic?

Basically, I am suppose to check if $f(x)=f(x+T)$. however my function is a bit complex: $x(t)=10\cos(20000\pi t)+0.5\cos(24000\pi t)+0.5\cos(16000\pi t)$ How shall I check if this function is ...
2
votes
2answers
55 views

“Slow” and “fast” rates of convergence

I have recently read about convergence and divergence. However, I am having trouble understanding how something can converge/diverge "slowly" or "fast". If you sum up two series (that converge to the ...
0
votes
0answers
44 views

If a series converges does its sequence of partial sums converge?

By Definition, a series $\sum_{n=1}^\infty a_n$ converges if it's sequence of partial sums $S_n = \sum_{k=1}^n a_k$ converges. My question is, is the converse true? If $\sum_{n=1}^\infty a_n$ ...
1
vote
1answer
48 views

One-sided Derivative Question

Let's say we define $$D_{+}f(x):=\lim_{h\to 0^+}\frac{f(x+2h)-f(x+h)}{h}$$ to be the "right-handed" derivative. This way the function does not have to exist (or equal what it 'should') at the point ...
0
votes
0answers
36 views

Name of this Formula [Spherical Earth projected to a plane]

I am using a formula to calculate the distance between two coordinates. Basically this is the Pythagorean theorem. I saw this formula on Wikipedia and it works perfectly for my use case. However I ...
0
votes
1answer
50 views

Is there a name for this theorem about the convergence of a function?

Let $f(x)$ be a continuous function over $\mathbb{R}$ such that for all $a < b$, we have $a < f(a) < b$. Then, for any $x < b$, the sequence $\{t_n\}$ defined by $t_0 = x, t_n = ...
5
votes
1answer
97 views

the word “derivative”

When did the word "derivative" come into use in calculus, and why? As in Can the word "derive" be used to mean "take the derivative of"? the word "derivative" in normal English ...
1
vote
2answers
92 views

Are the differential and derivative of a single-variable function exactly the same thing?

I just started taking a calculus class but I got in late and it had already started like weeks ago, so I'm completely lost. I believe the teacher uses this same formula in order to get the ...
8
votes
6answers
2k views

What is the difference between an indefinite integral and an antiderivative?

I thought these were different words for the same thing, but it seems I am wrong. Help.
0
votes
0answers
21 views

Definition of Range as Minimal Interval Containing Codomain

I am studying continuous functions where the domain is some interval (which may or may not be bounded, closed, etc). I am thinking about how continuity is related to other function properties, ...
2
votes
1answer
120 views

Meaning of 'small real parameter'?

Consider the family of functions on $[a,b]$ given by $y(x; \epsilon) : = u(x) + \epsilon \eta (x)$, where the functions $\eta = \eta (x)$ are twice continuously differentiable and satisfy $\eta(a) ...
1
vote
1answer
33 views

The term “maximal solution” for PDE

A solution $x(t)$ of the ODE is called maximal if it is defined on an open interval and cannot be extended to any larger open interval. from "Ordinary Differential Equation". Alexander ...
4
votes
0answers
2k views

What's the difference between early transcendentals and late transcendentals? [closed]

Anton, Bivens, and Davis have a calculus book with late transcendentals and Stewart has a calculus book with early transcendentals. What's this all about? edit 1: (Both terms show up in the titles of ...
1
vote
1answer
255 views

What is a tail sequence?

The question is self-explanatory. What is a tail sequence or a tail of convergent sequence? Thanks
6
votes
4answers
124 views

The limit of $f$ or the limit of $f(x)$?

I have read before that $f$ denotes the function $f$ whilst $f(x)$ denotes the value of the function $f$ at $x$. What is right? To say that the limit of $f$ as $x$ tends to $a$ is $L$ or to say that ...
2
votes
1answer
171 views

Bounded open interval

This is just a quick terminology question. My textbook is talking about the continuity of $ f $ over a bounded open interval $ (a,b) $. Am I right in assuming that this means $a$ and $b$ are finite? ...
7
votes
6answers
919 views

What is the meaning of infinitesimal?

I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ...
12
votes
3answers
230 views

Why “integralis” over “summatorius”?

It is written that Johann Bernoulli suggested to Leibniz that he (Leibniz) change the name of his calculus from "calculus summatorius" to "calculus integralis", but I cannot find their correspondence ...
1
vote
5answers
826 views

Define Onto and one to one meaning

i understand what one to one means. However, im struggling with understanding of onto Can anyone give me an example of onto? i want to understand what onto means.Can anyone explain what onto means in ...
2
votes
2answers
56 views

Is it ever proper to say that the limit of a function equals infinity?

If I calculate a limit and get the value $\infty$, what is the proper way to communicate this? Can I say that the $\lim_{n\to\infty}a_n=\infty$ and therefore the sequence $\{a_n\}$ diverges, or do I ...
4
votes
1answer
1k views

Difference between functional and function.

I have come across the term 'functional'. How is a 'functional' different from a 'function'? The exact term I came across was 'statistical functional.' In terms of the background, can you please ...
3
votes
2answers
140 views

Does “indeterminate” mean “divergent”?

Just learning about series and someone tried to tell me that when doing the alternating series test, if the limit is indeterminate, it means it is divergent, and I wanted to know what exactly the ...
2
votes
1answer
178 views

What is the purpose of defining the notion of inflection point?

What is the purpose of defining inflection point? I know that it is defined to be the point where the second derivative is zero and the second derivative sign changes. It has to have some purpose ...
4
votes
3answers
282 views

What does the phrase “except possibly” mean?

This term was used by my math professor when he was teaching us limits. This term also appears online like this "Let f be a function which is defined on some open interval containing $a$ except ...
7
votes
2answers
4k views

Difference between “undefined” and “does not exist”

What is the difference between the terms "undefined" and "does not exist", especially in the context of differential calculus? Most calculus materials state, for example, that $\frac{d}{dx}{|x|}$ ...
0
votes
2answers
102 views

smoothness of multivariable functions

What does it mean for a function $\mathbb{R}^n \to \mathbb{R}^m$ to be smooth? I see this in books, but typically we only talk about smoothness when the target set is $\mathbb{R}$.
4
votes
0answers
210 views

What does “toy-contour” mean?

When I reading Complex Analysis written by Elias M. Stein. In Chapter 2, he had introduced a notion "toy contour "without explaining. what does this exactly mean?
4
votes
2answers
774 views

Slope of a nonlinear curve at a single point

This part of my microeconomics lesson plan has me baffled. Consider for example the nonlinear continuous and differentiable function Y = f(X) = X 2 + 4. Suppose we want to know its slope at the ...
2
votes
2answers
955 views

Is the tangent function (like in trig) and tangent lines the same?

So, a 45 degree angle in the unit circle has a tan value of 1. Does that mean the slope of a tangent line from that point is also 1? Or is something different entirely?
1
vote
1answer
57 views

Properly say superscripts locations

In english: $\bigl(x+y^2\bigr)$ is ('x' plus 'y' squared) $(x+y)^2$ is ('x' plus 'y' squared) How can I make the difference in english between the two?
5
votes
1answer
103 views

What do you call a function differentiated with respect to all of its arguments?

Just a simple question. Let $f(x_1, x_2, \ldots, x_n)$ be a smooth function. Is there a particular name for the function $$\frac{\partial^n f}{\partial x_1 \, \partial x_2 \cdots \partial x_n}$$
7
votes
2answers
2k views

Difference between Slope and Gradient

It has been a few years since studying contour maps. Often I hear slope and gradient interchangeably in describing steepness. Does anyone know any good definitions and analogies of slope and ...
6
votes
2answers
295 views

Why does Maclaurin get his own polynomial?

Why is a Taylor polynomial centered around $0$ called a Maclaurin polynomial? It's only a special case of the Taylor polynomial, and it is calculated the exact same way as a Taylor polynomial centered ...
7
votes
3answers
623 views

What is the importance of functions of class $C^k$?

In all calculus textbooks, after the part about successive derivatives, the $C^k$ class of functions is defined. The definition says : A function is of class $C^k$ if it is differentiable $k$ ...
1
vote
3answers
639 views

What does “increases in proportion to” mean?

I came across a multiple choice problem where a function $f(x) = \frac{x^2 - 1}{x+1} - x$ is given. One has to choose the statement that is correct about the function. The different statements about ...
3
votes
2answers
2k views

How to explain what it means to say a function is “defined” on an interval?

I am having difficulty in explaining the terminology "defined" to the students I am assisting. Here is the sentence: If a real-valued function $f$ is defined and continuous on the closed interval ...
0
votes
0answers
74 views

“componentwise constant”?

This is a trivial vocabulary question. It seems to me that "constant on every connected component of the domain" would be a reasonable definition of the term "componentwise constant", provided that ...
6
votes
3answers
5k views

Meaning of the phrase “Limit does not exist”.

I read in Stewart "single variable calculus" page 83 that the limit $$\lim_{x\to 0}{1/x^2}$$ does not exist. How precise is this statement knowing that this limit is $\infty$?. I thought saying the ...
6
votes
2answers
220 views

terminology: what is meant if someone writes “calculus of ..”?

This question might be a little soft as it does not have a definite answer, so I hope I do not break the conventions of this forum by posting it here. I have now come across the term "calculus of ...
1
vote
3answers
2k views

why do we use 'non-increasing' instead of decreasing?

In english based math language it seems that non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing) decreasing $\Longleftrightarrow$ strict less ( strict decreasing) ...
1
vote
2answers
314 views

Is there actually a difference when we say 'perpendicular' vs 'tangent'

I find myself often getting these two words mixed up a lot. So let's say I have a simple graph of $y = t^2$ and $x = t$. If a line is tangent to the curve at the origin, it would only be the line y ...
4
votes
3answers
741 views

Can the word “derive” be used to mean “take the derivative of”?

Back when I was in high school, the usage of the word "derive" to mean "take the derivative of" was really widespread. It always bothered me because I felt that the proper verb should be ...
4
votes
1answer
251 views

If $f''(x)=0$ but is not an inflection point, what is it called?

If the second derivative of a function $f(x)$ equals zero at point $x_0$ ( $f''(x_0)=0$ ), the point is an inflection point if the concavity changes. Here's an example of an inflection point. ...
4
votes
2answers
334 views

Name of property describing the number of times a function changes concavity?

For example, $f(x)=\sin x$ changes concavity an infinite number of times, $f(x)=x^3-x$ has two regions of concavity (changing concavity once), and $f(x)=x$ changes $0$ times. Is there a name for ...