1
vote
2answers
27 views

Can this be written in standard “vector calculus notation”?

A formula for the gradient of the magnitude of a vector field $\mathbf{f}(x, y, z)$ is: $$\nabla \|\mathbf{f}\| = \left(\frac{\mathbf{f}}{\|\mathbf{f}\|} \cdot \frac{\partial \mathbf{f}}{\partial x}, ...
0
votes
2answers
79 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
3answers
60 views

whats the difference between $|v|$ and $||v||$?

$v$ being a vector. I never understood what they mean and haven't found online resources. Just a quick question. Thought it was absolute and magnitude respectively when regarding vectors. need ...
1
vote
2answers
50 views

row vs. column vector derivative

I am reviewing calculus from Spivak's Calculus on Manifolds, and it looks like he is being (very?) cavalier with regards to when vectors should be written as columns or rows. Let $f:\mathbf{R}^n \to ...
0
votes
1answer
47 views

Implicit function theorem notation question

Given a function $F:\mathbb{R}^2\to\mathbb{R}$, by the implicit function theorem, the relation $F(x,y) = 0$, defines $y$ implicitly as function of $x$ in a neirbourhood of $(x_0,y_0)$, provided ...
1
vote
1answer
56 views

Gradient notation - understanding subscripts

In one of my textbooks the following notation is used to describe strain components in a displacement field: $$\begin{bmatrix} S_1 \\ S_2 \\ S_3 \\ S_4 \\ S_5 \\ S_6 \end{bmatrix} = \begin{bmatrix} ...
1
vote
1answer
54 views

Explanation of this vector notation?

In a journal article I'm reading, a vector is written as: \begin{gather*} R_{ON} = \Sigma_{\varphi_{k}} \langle x_{\varphi_{k}} \rangle_{ON} \cdot e^{j \varphi_{k}} \end{gather*} What's really ...
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: ...
0
votes
1answer
134 views

How to write the expression with multi-index notation?

I need some help for writing the following expression with multi-index notation, $$\sum_{i_1, \ldots, i_p=1}^n \frac{\partial^{2p}}{\partial x_{i_1}^2\ldots \partial x_{i_p}^2}f(x, \xi),$$ where ...
2
votes
6answers
74 views

Why $\vec{r}$ is commonly use for vector equation?

I'm wondering why $\vec{r}$ is commonly use in mathematics (vector calculus, line integrals) and physics for denote the vector equation. Edit/Added clarification: I'm wondering why the letter $r$ is ...
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 ...
0
votes
1answer
48 views

Notation question arising from physical problem

I'm a physicist and am a bit confused about the notation in one of the computations here: Let $S = S(U,V,N)$ be a real valued scalar function, and $z = (U,V,N)$. Let $\lambda \in \mathbb{R}$. Given ...
0
votes
1answer
128 views

What is this notation relating to the Jacobian matrix operation?

In the below image, in the very bottom-most equation is the partial differential at the end of the equation being multiplied to every element in the inverse Jacobian matrix (and then beta_n added)? Or ...
3
votes
1answer
76 views

Simplifing formulas using tensor notation

Im trying to symplify formulas like: $$\operatorname{div}(\operatorname{rot}\vec{F}),\qquad \operatorname{rot}(\operatorname{rot}\vec{F}) $$ or something more strange like: ...
1
vote
1answer
60 views

Is this notation for Stokes' theorem?

I'm trying to figure out what $\iint_R \nabla\times\vec{F}\cdot d\textbf{S}$ means. I have a feeling that it has something to do with the classical Stokes' theorem. The Stokes' theorem that I have ...
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 ...
1
vote
1answer
275 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 ...
5
votes
1answer
84 views

Confusion over calculus notation (differentials/derivatives)

I have read from multiple sources that dy/dx is not to be interpreted as a ratio as the idea of 'dy' and 'dx' themselves will lead to logical difficulties. However, I have seen in many areas (e.g. ...
2
votes
1answer
100 views

Understanding some differential notations…

What this notation mean? (I know that is a partial derivative, but I don't understand the meaning of the evaluation bar at the right) $$\frac{\partial g}{\partial T}\Big|_{SA,p}$$ Is this relation ...
1
vote
1answer
54 views

How do I write the integral over all $x$ in $\Bbb R^n$?

If I have $f:\mathbb{R}^n\to\mathbb{R}$ I would write the integral over some region $\mathcal{R}\subset\mathbb{R}^n$ like: $$ \int_\mathcal{R}f(\mathbb{x})\mathrm{d}\mathbb{x}. $$ What subscript ...
1
vote
2answers
1k views

What does $C[0,1]$ mean?

In the context of real analysis, I have found this question: For each $$f \in C[0,1] $$ there is a series of even polynomials , which converge uniformly on $[0,1]$ to f. What is $C[0,1]$ ? Is it ...
0
votes
1answer
44 views

Given a function space with a norm , what is the meaning of writing $||.||$ when the used norm is $||.||_\infty$

Example 1 Given $$C_{0}(\mathbb{R}^{n})=\{f\in C(\mathbb{R^n} \ | \ \ \exists R \ge 0 \ \text{such that } f(x)=0 \ \text{for} \ ||x||\ge R \}$$ and $$||f(x)||_{\infty} = \max_{x\in R^n}|f(x)| $$ ...
4
votes
1answer
417 views

Writing “$\nabla f$” or “$\operatorname{grad} f$”

When hand-writing the gradient of $f$ as "$\nabla f$" or "grad $f$", is it necessary to indicate that it is a vector using the usual vector markings (cap, arrow, wavy line, etc.)?
0
votes
1answer
433 views

Understanding the notation of the gradient of a vector function

In my finite element method book, there is a notation which is confusing me. Given $v:R^2\rightarrow R^2$, I'm supposed to evaluate $\sigma\cdot \nabla v^T$ where $\sigma$ is a smooth tensor ...
1
vote
0answers
205 views

Understanding Line integral notation

I'm evaluating a line integral written in the form: $\int_{\partial\Omega_1} v\nabla u\cdot n$ where $\partial \Omega_1$ is simple curve forming one part of the boundary $\partial\Omega$ of a ...
1
vote
2answers
284 views

Explain the Stokes -theorem from differential from into Integral form

I want to understand the Stokes -theorems deeper. I am trying to understand the operation from $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$ to ...
1
vote
2answers
338 views

What is the difference between $d$ and $\partial$?

After seeing the following equation in a lecture about tensor analysis, I became confused. $$ \frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds} $$ What exactly is the difference ...
3
votes
1answer
355 views

Notation in Munkres' $\textit{Analysis on Manifolds}$

I am trying to understand Theorem 9.1 of 1991 copy of Munkres' Analysis on Manifolds. I have stated what I don't understand below; there is a heading in bold. This theorem is a precursor to the ...
2
votes
4answers
977 views

Second order partial derivatives - notation

I have seen both of these used, and people around me seem to disagree, so which one is correct: (first derivative with respect to x, then y): (1) $$\frac{\partial }{\partial y}(\frac{\partial ...
3
votes
2answers
123 views

What is the difference between these two derivative expressions?

is there a difference between $\frac{\partial^2 }{\partial x^2}$ and $(\frac{\partial }{\partial x})^{2}$? I have to tell if a differential equation is linear, and $(\frac{\partial }{\partial x})^{2}$ ...
2
votes
2answers
154 views

Using nabla with partial derivatives and the Laplace operation $\partial_x^2+\partial_y^2+\partial_z^2$

Source of the problem p.812 here. Suppose $$\bar{F}(x,y,z)=(xy-z^2)\bar{i}+(xyz)\bar{j}+(x-y^2-z^2)\bar{k}.$$ I am concerned where I need to nabla an unit vector for example with $$\triangledown ...
4
votes
2answers
1k views

How do you pronounce (partial) derivatives?

I am not an English speaker that is why I asked this question. In addition, I think english.stackexchange.com is not the proper place to ask this because (I am so sorry) I don't think most of them ...
4
votes
1answer
107 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 ...
3
votes
1answer
278 views

What does this notation mean? $\frac{\partial f}{\partial x}(x+y)=\frac{\partial }{\partial x}f(x+y)$

$$\frac{\partial f}{\partial x}(x+y)=\frac{\partial }{\partial x}f(x+y)$$ I was just wondering what the left-hand side mean. (or how to do the operation based on the notation of the LHS, given a ...
0
votes
0answers
178 views

differential notation with del operator

For any state function of n independant variables $F(x_1, \dots ,x_n)$, $$dF(x_1, \ldots ,x_n)=\frac{\partial F}{\partial x_1} dx_1 + \cdots + \frac{\partial F}{\partial x_n} dx_n$$ where each partial ...
3
votes
3answers
187 views

Notations about multiple integrals

I have the following sum: $$ S = \int f_1(x_1) dx_1 + \int f_2(x_2) dx_2 + \int f_3(x_3) dx_3 $$ Letting $x = (x_1,x_2, x_3)$ and $f = (f_1(x_1), f_2(x_2), f_3(x_3))$ can I rewrite $$ S = \int ...
2
votes
1answer
150 views

Bound for multi-index sum

I have difficulties in evaluating the multi-index notation in the following context: Let $x \in R^n$ and let $i$ be a multi-index, $i=(i_1, \dots, i_n)$. Now I want to know the bound of the sum ...
0
votes
0answers
191 views

Expression/unknown Notation for Taylor expansion in multivariate case

For a $C^3$-function $f:R^n \to R$, $x \mapsto f(x)$ I have the Taylor series $$ f(x+y)=f(x) + \nabla f (x)^T y + \frac{1}{2} y^T (\nabla^2 f(x)) y + \sum_{|j|=3} \frac{D^j f(z)}{j!} y^{j}$$ with ...
1
vote
1answer
173 views

Notation for Curve/Path Concatenation in Calculus Integrals

I can't find this online from a simple search, and I cannot remember. Given two curves/path $C$ and $D$, what is the notation for path concatenation when describing a path integral? Here are some ...
10
votes
1answer
265 views

What exactly does $\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$ refer to?

I have been asking a rather few questions of this nature lately, maybe I'm starting to realise math notation isn't as uniform as I initially thought it would be... Question: Does this notation ...
4
votes
1answer
248 views

Notation - Two adjacent vectors?

I'm studying multivariable calculus at the moment and have come across equations involving two bolded variables placed side by side, like so: $$ \nabla \mathbf{f}=\frac{\partial {{f}_{j}}}{\partial ...
1
vote
2answers
345 views

A question on notation: What does $\nabla |\overrightarrow{a} \times \overrightarrow{r}|^n$ mean?

I sort of asked a version of this question before and it was unclear; try I will now to make an honest attempt to state everything clerly. I am trying to evaluate the following, namely $\nabla w = ...