0
votes
1answer
17 views

Notation regarding the continuity equation for conservation of mass

I have the following equation for the net mass flow out of a control volume through a surface $S$ - $$\int \int_S p \overrightarrow{V} \cdot \overrightarrow{d}S$$ (Actually there should be an ellipse ...
3
votes
2answers
80 views

Weird integral symbol : $\mathrel{\int\!\!\!\!\!-}$

What does this integral sign mean ($\int$ with line going through the middle)? $$ \mathrel{\int\!\!\!\!\!\!-} $$ (It had something to do with the Beckenbach-Radó Theorem)
0
votes
3answers
54 views

Find the value of $m$ given that $\displaystyle\int_0^m \dfrac{dx}{3x+1}=1$

I have to find the value of $m$ such that: $\displaystyle\int_0^m \dfrac{dx}{3x+1}=1.$ I'm not sure how to integrate when dx is in the numerator. What do I do? edit: I believe there was a typo in ...
9
votes
2answers
530 views

Why do we need the absolute value signs when integrating the square of a function?

Why do we need the absolute value signs in the definition of square-integrable function? As seen below: $$ \int_{-\infty}^{\infty} \lvert f(x) \rvert^2 dx < \infty $$
0
votes
1answer
43 views

Integral notation

I have encountered the following integral: $\int_{x-d}^{x+d}f(y)dy$ I am trying to figure out what is the role of $d$ in this integral. Is the $d$ at the beginning of the integral the same as the ...
2
votes
3answers
204 views

Why are variables in integration by substitution so counter intuitive?

Integration by substitution is defined as something like $\displaystyle\int_a^b f(\phi(t))\phi'(t)dt = \int_{\phi(a)}^{\phi(b)} f(x)dx$ But for my taste, the variables $x$ and $t$ are exactly ...
4
votes
1answer
111 views

How do you write an integral and why

A. Year 1 Calculus Student Approach $$ F(x) = \int f(x') dx\, $$ B. Random math paper you find online approach $$ F(x) = \int dx f(x') \, $$ C. Spivak $$ F(x) = \int f(x) \, $$ D. ??? (Edit) ...
4
votes
4answers
762 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
1
vote
2answers
70 views

Notation for Surface Integral in $\mathbb{R}^3$

Recently, a paper of mine got accepted, but the reviewers are struggling with the (in my view) standard notation for surface integrals in $\mathbb{R}^3$: Let $\Gamma \subset \mathbb{R}^3$ be a ...
1
vote
1answer
47 views

Integral notation from cartesian from polar coordinates

Given an integral $$I=\int\limits_{\mathbb{R}^n} \cdot \; dx,$$ we can introduce polar coordinates, such that $$I=\int\limits_{\Bbb S^{n-1}} \cdot \; d\theta.$$ Another way to express the latter one ...
1
vote
0answers
39 views

Integration notation

could someone explain the following notation: $u(x,t):=\int^t_0 v(x,t:\tau)d \tau$ It's come up but I don't understand how to interpret the semicoloned tau
1
vote
4answers
75 views

Integrating a term again and again

So if you have $f''(x) = 24x$ you know you want to integrate it, because it would look much better integrated, so now we have $f'(x) = 12x^2$, but it could still look better, so we integrate it to ...
-1
votes
1answer
47 views

Strange Integral Notation? [duplicate]

When reading about an certain algorithm (about parameter estimation for Kalman Filtering page 7 eq 57) I found this notation: $\int dx f(x)$ which is normally written as $\int f(x) dx$. I spent a ...
5
votes
1answer
78 views

Intuition for Integration of Differential Forms

In mathematics, we define $dx^i$ as linear functionals, when speaking of integration. However, in physics, we interpret $dx^i$ as very small quantities. There is nothing inherently small about a ...
20
votes
6answers
519 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 ...
0
votes
2answers
92 views

Double integral notation

Over a region D (a bounded, closed and connected region), can we write the double integral $\iint\limits_D \, f(x,y)\,dx\,dy$ as $\iint\limits_D \, f(x,y)\,dy\,dx$ (note the order of $dx$ and $dy$)?
3
votes
2answers
268 views

How is it that treating Leibniz notation as a fraction is fundamentally incorrect but at the same time useful?

I have long struggled with the idea of Leibniz notation and the way it is used, especially in integration. These threads discuss why treating Leibniz notation as a fraction and cancelling ...
5
votes
2answers
94 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 ...
4
votes
1answer
69 views

Why $dt/t$ in Mellin transform

I've noticed that often when people write the Gamma function $\Gamma(s) = \int_0^\infty t^{s-1}e^{-t}\,dt$, that they write it like $$ \Gamma(s) = \int_0^\infty t^s e^{-t}\,\frac{dt}{t} , $$ where ...
1
vote
3answers
43 views

Notation of this set in a set?

I am currently struggeling with the following notation: For $\epsilon \in (0,1)$ and $p \in (0,\infty)$, consider the following subset of $L ^p$: $M(p,\epsilon)=\{f \in L^p:m \{x:|f(x)| \ge \epsilon ...
2
votes
0answers
65 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
0answers
83 views

Integration by parts and notation.

I've just found that I must be missing something about the integral notation when it comes to the integration by parts. First, $\int_a^b \! f(x) \, \mathrm{d}x$ is perfectly clear: $\mathrm{d}x$ is ...
2
votes
1answer
43 views

A notational problem of a double integral

What does the following double integral mean?I don't quite understand the notation. $$\int_0^1dy\int_\sqrt{y}^\sqrt{2-y^2}f(x,y)~dx~.$$ Does it equal to ...
0
votes
3answers
88 views

Prime notation in integral?

Recall the definition of potential energy: $$ U_x-U_{x_0} = -\int^x_{x_0}F_x(x')dx' $$ I've seen the integral definition of work, but not this - the thing I'm specifically interested in is the ...
2
votes
1answer
185 views

How can a single integral equal a triple integral?

Here is part of a discussion about the gravitational potential of a sphere: Let $dx$ $dy$ $dz$ represent an infinitesimal volume containing matter of density $\rho$ and mass $dm$. Then the ...
0
votes
1answer
50 views

Integrals with infinite bounds sometimes written as limits, sometimes not?

When I saw Wikipedia's notation for the inverse Laplace transform, I became curious if there was a reason behind it. Is there a reason why Wikipedia writes the inverse Laplace transform as this ...
6
votes
1answer
274 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: ...
5
votes
1answer
100 views

Meaning of the following integal

What does $\int d^3 x $ mean? I found this in a lecture on quantum field theory, and it was not explained.
2
votes
0answers
66 views

Why write $\mathrm dx, \mathrm dt$ etc. at the beginning of an integral? [duplicate]

I've noticed that many people here (on Math.SE) as well as elsewhere write integrals out like this: $$\int^a_b \mathrm dt \; f(t)$$ instead of the more common (at least from what I've seen): ...
6
votes
4answers
315 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?
10
votes
3answers
334 views

Meaning of $\int\mathop{}\!\mathrm{d}^4x$

What the following formula mean? $$\int\mathop{}\!\mathrm{d}^4x$$ I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following ...
9
votes
3answers
342 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 ...
2
votes
1answer
406 views

Integral sign with circle (AND arrow on the circle) through it

I know from multivariable calculus that the integral sign with circle in its middle means integrating along a closed path. So when I encountered in complex analysis the above integral sign but with ...
0
votes
3answers
75 views

Different methods to write an integral

I saw someone write this for showing substitution. Is it correct. $$\int \frac{2x}{\sqrt{x^2+2}}\, \mathrm{d}x$$ $$\int \frac{\mathrm{d}u}{\sqrt{u}}\, \mathrm{d}\left(2xdx\right)$$ Just wondering ...
0
votes
2answers
200 views

Evaluating using BRA KET Notation?

Evaluate $\langle 0 \mid x^3 \mid 1\rangle$, assuming that all the wave functions you encounter are normalized eigenfunctions of the harmonic oscillator Hamiltonian, without Mathematica, Maple, or ...
11
votes
4answers
846 views

Why doesn't Spivak ever write $dx$ in an integral?

I've noticed that Spivak, and many other analysis books I read like Munkres, do not use $dx$ when they integrate. Why is that? This is a serious question.
5
votes
5answers
1k views

What is this symbol $\int$ called?

Title says it all (I think). I'm betting it's something to do with Standard Integrals though. What is this symbol "$\int$" called?
3
votes
1answer
234 views

Integration Antiderivative vertical bar [duplicate]

Possible Duplicate: What is the name of the vertical bar? When taking a definite integral, the first step is finding the anti-derivative. Once you have gone through all the steps to ...
1
vote
3answers
427 views

What does $\inf\int$ mean?

What does notation $$\inf\int f(x) \,\mathrm{d}x$$ stand for? I noticed it in a question on this site. Any keywords, links or stories about this or similar notations will be appreciated :) Sorry ...
5
votes
0answers
236 views

Is there a name or definition for this popular notation?

I'm sorry if this is a silly question. I've done quite a bit of searching and have not found any definition or name for this symbol/usage, despite immense popularity and convenience. The sources I've ...
0
votes
1answer
143 views

“Discovering” the indefinite integral's notation

I'm currently reading Keisler's Elementary Calculus -- An Infinitesimal Approach, which develops the main results usually thought in undergrad calculus using Robinson's hyperreal numbers (instead of ...
2
votes
1answer
114 views

Limit of an n-ary product

Since a definite integral is defined as $$\lim_{n\to\infty} \sum_{i=0}^n f(x_i^*)\,\Delta x = \int_a^b f(x)\,dx$$ and the integral is much easier to calcluate than a sum, if we change the sum to a ...
5
votes
1answer
294 views

$\int f(x) dx $ is appearing as $\int dx f(x)$. Why? [duplicate]

A few of us over on MITx have noticed that $\int f(x) dx $ is appearing as $\int dx f(x)$. It's not the maths of it that worries me. It's just I recently read a justification (analytical?) of the ...
3
votes
1answer
101 views

Confusion with (notation in) a surface integral problem.

I encountered the following problem in a practice Advanced Calc exam, and I have an issue. Suppose $\phi:\mathbb{R}^3\to\mathbb{R}$ is a strictly positive function satisfying ...
0
votes
2answers
264 views

Notation for antiderivative

I came across the following: $$ F(x) = \int x^3 \cos(x)dx $$ where $F$ is understood to be a primitive of $x^3 \cos(x)$. I find this confusing, because of the "same" $x$ appearing on both sides of ...
1
vote
0answers
76 views

Notation for application of the sequence of integrals

In the paper I'm writing I often encounter expressions like $$\int dx_1 \int dx_2 \ldots \int dx_N \mathrm{<something>}.$$ In this form it is not that cumbersome, but things get worse when ...
4
votes
2answers
506 views

Necessity of Brackets for Integration

Suppose I want to integrate $f(x)+g(x)$. Can this be written as $\int f(x)+g(x)\, dx$ or are brackets necessary, i.e. $\int \left(f(x)+g(x)\right) \,dx$?
1
vote
1answer
511 views

This multiple integral notation, has it got a name? $\int dx \int dy \, f(y,x)$

I've encountered, on Wikipedia (examples below), an integration notation which seems to be prefix-style: the integral sign is immediately followed by the $\mathrm dx$ (or $\mathrm dy$, or what have ...
3
votes
3answers
349 views

Notation pedantry (integration by substitution)?

In a summative assessment, I lost a mark due to this: $$f_X(x)=\frac{\lambda}{\sqrt{2\pi}}\int_{-\infty}^\infty y^2\exp\left\{-\left(\frac{1}{2}+\lambda x\right)y^2\right\} \; dy$$ Now let ...
4
votes
1answer
112 views

Notation for some integrals

I've seen some problems where the OP writes integrals in this form $$\int {dt} f\left( t \right)$$ or for double integrals $$\int {dx} \int {dtf\left( {t,x} \right)} $$ Do they represent another ...