4
votes
1answer
40 views

Uncomfortable using Leibniz notation for the chain rule.

I am working through the following solved problem which uses separation of variables to get two ODEs. The problem is to show that $$\frac{1}{\sin\theta ...
16
votes
6answers
410 views

The formalism behind integration by substitution

When you are doing an integration by substitution you do the following working. $$\begin{align*} u&=f(x)\\ \Rightarrow\frac{du}{dx}&=f^{\prime}(x)\\ \Rightarrow ...
1
vote
0answers
26 views

Notation for near optimal solution

Usually, $x^*$ is used to denote the optimal solution to a maximization problem. I need a notation to describe a solution that is not optimal but "good enough." In my case this solution is the first ...
2
votes
2answers
61 views

Taylor series of $\sqrt{1+x}$ using sigma notation

I want help in writing Taylor series of $\sqrt{1+x}$ using sigma notation I got till $1+\frac{x}{2}-\frac{x^2}{8}+\frac{x^3}{16}-\frac{5x^4}{128}+\ldots$ and so on. But I don't know what will come in ...
1
vote
4answers
59 views

Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is ...
2
votes
1answer
53 views

What does this notation involving greater than sign mean?

Basically it says that $$|x|>1,2n|a_{n-1}|,\ldots,2n|a_0|$$ Does this mean that $$|x|>\max(1,2n|a_{n-1}|,\ldots,2n|a_0|)$$ It seems to be essential when proving that polynomial whose degree ...
5
votes
2answers
83 views

is there any history at all for this notation of partial anti-derivatives?

i have searched but can not find examples of any published book or online articles that use this notation: $$\int f(x,y) \partial x$$ seems it would be useful for example here: $$\int_I\int_J ...
1
vote
4answers
64 views

Surprised over notation in fundamental theorem

So I'm looking into the fundamental theorem of calculus and I'm a bit weirded out by the notation used in part two. $$ \int_{a}^b f(t) dt = G(B)-G(a)$$ Why don't we say $F(b)-F(a)$? We are just ...
3
votes
2answers
127 views

Usage of $\cdot$ in calculus

I often find myself caught in the dilemma of whether or not to use the symbol $\cdot$ in calculus. Take for example, the chain rule: $$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$$ Is the ...
2
votes
3answers
83 views

Why are there so many notations for differentiation?

There are so many notations for differentiation. Some of them are: $$ f^\prime(x) \qquad \frac{d}{dx}(f(x))\qquad \frac{dy}{dx}\qquad \frac{df}{dx}\qquad D f(x)\qquad y^\prime\qquad D_x f(x) $$ Why ...
1
vote
1answer
31 views

Shorthand method for expressing the limit of something

Solving limits takes a lot of steps sometimes, but I feel bad leaving out the limit each time I do something and rewrite "=". Is there a shorthand method for writing the limit? $$\lim_{x \to p}f(x) = ...
1
vote
1answer
73 views

Simplification of a nested sum

I have a nested sum like so: $$\underbrace{\sum_{k_1=k_0}^{k^*} \ ... \sum_{k_n=k_{n-1}}^{k^*}}_{\text{n times}} 1\quad\ \text{with}\ \ n, k_0, k^* \in \mathbb{N},\ k^*\geq k_0$$ Is there a general, ...
1
vote
1answer
38 views

Seperation of variables justification?

I haven't found a similar question on Math SE, but I may not have looked enough because I find it hard to believe someone hasn't already asked this. Anyways, here goes: I'm studying mathematics, but ...
2
votes
0answers
60 views

Integrating With Respect To $x$

Suppose I have the first derivative of the function $y$, $\displaystyle \frac{dy}{dx} = g(x)$. Furthermore, suppose I want to obtain the function $y$ by integrating with respect to $x$: ...
1
vote
3answers
113 views

Equation with the big O notation

How I can prove equality below? $$ \frac{1}{1 + O(n^{-1})} = 1 + O({n^{-1}}), $$ where $n \in \mathbb{N}$ and we are considering situation when $n \to \infty$. It is clearly that it is true. But I ...
1
vote
0answers
31 views

Do we need to pay attention to the codomain of a differentiable function?

I came across the following definitions: We call $M\subset \mathbb R^N$ $m$-dimensional $C^k$-submanifold of $\mathbb R^N$ if for all $a\in M$ there is an open neighborhood $U$ of $0$ in $\mathbb ...
0
votes
1answer
48 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 = ...
0
votes
1answer
48 views

What is meant by this notation?

I found the following question: Let $\large f(x)=e^{x^2}$. Find and simplify $\large{f^{(3)}(x)}$. Is it asking to find the third derivative of the function? Which rule is used to find the ...
0
votes
1answer
80 views

Notation: using dot instead of argument

Is there any difference between: $f\left(\cdot , \theta \right)$ is continuous in x for each $\theta$ $f\left(x , \theta \right)$ is continuous in x for each $\theta$
2
votes
1answer
51 views

Partial differentials vs normal differential (notation question/clarification only)

In physics, it seems like the use of $\dfrac{dy}{dx}$ and $\dfrac{\partial y}{\partial x}$ are used somewhat interchangeably. My understanding is that, technically $\dfrac{dy}{dx}$ is only ...
0
votes
1answer
53 views

Notation for derivative of a 2 argument function w.r.t its second argument

For functions of one argument, the "Newton" notation for the derivative of that function is concise and unambiguous. For example, if I want to express $$ \lim_{h\to 0} \frac {f(x^2 + h) - f(x^2)}{h} ...
1
vote
2answers
60 views

What does this function mean?

$$f(x) = \frac{x}{e^{x^2}}$$ Differentiate $f(x)$. How should the above function be interpreted? Is the function equivalent to: a)$$f(x) = \frac{x}{e^{x^2}} = \frac{x}{{(e^x)^2}} = ...
2
votes
2answers
212 views

About some notation of the derivative

I'm currently Rudin's Principles of mathematical analysis, there is this definition of the "partial derivative". If $f$ maps an open set $E\subseteq R^n$ into $R^m$ and $\{e_1,...,e_n\}$ and ...
1
vote
2answers
116 views

Notation regarding different derivatives

I am currently reading up on partial derivatives and differentials in general. And there are a few points that seem unlcear to me (notation-wise). For example, if $f:\mathbb R\to\mathbb R,x\mapsto ...
0
votes
1answer
69 views

Notation for differentiable

The conditions for the Mean value theorem is that if $f$ is defined on a closed interval $[a,b]$, f is continuous on [a,b] and differentiable on $(a,b).$ Then there exists a $\xi$ in [a,b] such that ...
0
votes
1answer
50 views

Question about notation in differential equations.

In general, an ordinary differential equation is in the form $$ \begin{cases} x'(t) = f(t, x(t)) \\ x(t_0) = x_0 \end{cases}. $$ When proving the existence and uniqueness theorems, an operator $T$ was ...
1
vote
1answer
41 views

Limit Comparison Test Defined entirely in symbolic notation

Is it possible to define the limit comparison test entirely with symbols (no textual explanation), or with as little textual explanation as possible? How? My latest best attempt: ...
2
votes
2answers
150 views

Does differentiation symbol need parentheses or?

Suppose I have this expression: $$\frac{d}{dx}(e^{x})^2 + 6$$ Does it mean to differentiate $6$ too or just the first term? This is an exercise on a calculus course that I'm doing on Coursera. ...
0
votes
1answer
104 views

Function and dependent variable are represented by the same symbol?

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$ , $y=y(t)$ where $t$ is the ...
1
vote
0answers
189 views

using the same symbol for dependent variable and function?

Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$, $y=y(t)$ where $t$ is the ...
2
votes
2answers
101 views

What exactly does the $d$ represent in $\frac{d}{dx}$?

When taking the derivative, such as $\frac{d}{dx}$, what exactly does the $d$ represent? The best answer so far is in for example $\frac{dy}{dx}$, the $d$ stands for change in and what follows the ...
0
votes
1answer
65 views

Intervals and sets

I never took calculus until now, but as a stat major I have sometimes used the notation $x\in [a, b]$ as an alternative way of writing $a\leq x \leq b$. Does it make sense to express it like this, or ...
4
votes
1answer
212 views

Derivative $\Delta x$ and $dx$ difference

This may seems like a dummy question but I need to ask it. Consider the definition of derivative: $$\frac{d}{dx}F(x) = \lim_{\Delta x->0}\frac{F(x+\Delta x) - F(x)}{\Delta x} = f(x)$$ Also: ...
1
vote
1answer
61 views

Show that a partial differential eqution is satisfied (is my notation okay?)

Let: $f:\mathbb{R}\rightarrow\mathbb{R}$ be differentiable. $u:\mathbb{R}^2\rightarrow\mathbb{R}$ and $u(x,y)=e^{-y}f(x+y^2)$ Show that: ...
4
votes
4answers
197 views

Leibniz notation for high-order derivatives

What is the reason for the positioning of the superscript $n$ in an $n$-order derivative $\frac{d^ny}{dx^n}$? Is it just a convention or does it have some mathematical meaning?
3
votes
0answers
55 views

partial derivative notation question

I'm reading a book called Correlated Data Analysis, Analytics, and Applications and I simply don't understand some notation. The author says, in chapter 2, page 26: A unit deviance is called ...
4
votes
1answer
107 views

Notation of derivatives…

I asked my teacher the difference between this notations. (1) $$\frac{dy}{dx}$$ (2) $$\frac{\delta y}{\delta x}$$ (3) $$\frac{\Delta y}{\Delta x}$$ He told me that there is no difference. I really ...
2
votes
3answers
67 views

Simple question about integrals?

simple question , if we define $\displaystyle F(x) = \int_a^x f(t) \, \mathrm{d}t$ does that mean $x>a$? or $x$ could be smaller than a even though that expressoin means it's an upper bound?, ...
1
vote
3answers
478 views

Why is the Jacobian matrix the transpose of what I would think it'd be/usefully be (total derivative is a synonym) (EDIT: I was a total wally)

I'm sorry this isn't a yes/no/am-I-right question but I seriously cannot see why the Jacobian/total derivative matrix is what it is? I am also using it as LaTeX practice (for maths) hence the barely ...
3
votes
6answers
277 views

What is $a$ in $F(x)=\int_a^xf(t)\,dt$?

I often see things like $$ F(x)=\int_a^xf(t)\,dt. $$ What is $a$? Is its value important? I ask this because I often get the feeling that $a$ could be any constant. I also see $a$ sometimes be ...
6
votes
4answers
309 views

Evaluating $\int \frac{dt}{(\cos(t))^2}$?

How do I solve an integral with a differential on top? E.g.: given this integral to evaluate: $$\;\int \frac{dt}{(\cos(t))^2}\;\;?$$ What does it even mean when there's a differential?
1
vote
0answers
135 views

What is meant by an integral with $\Omega$ as its range: $\int_{\Omega}$

What does it mean when $\Omega$ is range of integral? $$\iint G \frac{\partial U}{\partial n} - U(J k G) ds =\int_{\Omega}{G\left(\frac{\partial U}{\partial n} -j k U\right)R^2 dw} $$ I do not ...
2
votes
1answer
47 views

Notation for function being differentiable at a certain point

This question describes a notation for a function $f(x)$ being (continuously) differentiable on some domain $A$. Often, I see the requirement that some function $f(x)$ be differentiable only (or ...
2
votes
1answer
63 views

prime notation clarification

When I first learned calculus, I was taught that $'$ for derivatives was only a valid notation when used with function notation: $f'(x)$ or $g'(x)$, or when used with the coordinate variable $y$, as ...
0
votes
2answers
129 views

I don't understand this notation… - Series with ln

I found this notation in my book $$ \sum\limits_{i=1}^n \ln^n3 $$ and I don't know how to interpret it. Is it $$ \sum\limits_{i=1}^n \ln((1^n)\cdot3)\;? $$ And btw, how to check if this series ...
0
votes
1answer
77 views

question about epsilon, delta limit definition

Sometimes, when describing the closeness of $x$ to $a$ as being less than $\delta$, it's stated as $|x-a|<\delta$ and sometimes it's stated as $0<|x-a|<\delta$. What is the " $0<$ " part ...
1
vote
1answer
45 views

What does $\|$ mean in this definition of the total derivative of a function?

What does $\|$ mean in this definition of total derivative? picture:
8
votes
3answers
276 views

Understanding the differential $dx$ when doing $u$-substitution

I just finished taking my first year of calculus in college and I passed with an A. I don't think, however, that I ever really understood the entire $\frac{dy}{dx}$ notation (so I just focused on ...
1
vote
3answers
70 views

Double Integrals. Simple question, don't understand their wording.

Make a sketch of the region over which $$\int_0^{\pi/2} dx \int_0^{\sin(x)} dy$$ Would this be the same as $$\int_0^{\pi/2} 1 dx \int_0^{\sin(x)} 1 dy$$ Which simply evaluates to ...
2
votes
0answers
194 views

What's the difference between $\frac{\delta}{dt}$ and $\frac{d}{dt}$?

I have read the few questions on calculi notation, particularly the notations on partial and total derivatives. My question seems to have not been answered, or at least not brought to my attention. If ...