Tagged Questions
4
votes
1answer
40 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. ...
1
vote
1answer
45 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
109 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
32 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
397 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
311 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
93 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 ...
0
votes
2answers
194 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
166 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
272 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
557 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) $$\frac{\partial }{\partial y}(\frac{\partial ...
3
votes
1answer
93 views
What is the difference between these two derivative expressions?
is there a difference between $\frac{\partial^2 }{\partial x^2}$ and $(\frac{\partial }{\partial x})^{2}$? I have to tell if a differential equation is linear, and $(\frac{\partial }{\partial x})^{2}$ ...
2
votes
2answers
122 views
Using nabla with partial derivatives and the Laplace operation $\partial_x^2+\partial_y^2+\partial_z^2$
Source of the problem p.812 here. Suppose
$$\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
539 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 ...
3
votes
1answer
227 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
133 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
156 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
$$ S = \int ...
2
votes
1answer
118 views
Bound for multi-index sum
I have difficulties in evaluating the multi-index notation in the following context:
Let $x \in R^n$ and let $i$ be a multi-index, $i=(i_1, \dots, i_n)$.
Now I want to know the bound of the sum ...
0
votes
0answers
152 views
Expression/unknown Notation for Taylor expansion in multivariate case
For a $C^3$-function $f:R^n \to R$, $x \mapsto f(x)$ I have the Taylor series
$$
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}$$ with ...
1
vote
1answer
129 views
Notation for Curve/Path Concatenation in Calculus Integrals
I can't find this online from a simple search, and I cannot remember.
Given two curves/path $C$ and $D$, what is the notation for path concatenation when describing a path integral? Here are some ...
10
votes
1answer
194 views
What exactly does $\frac{\partial(y_1,\dots,y_m)}{\partial(x_1,\dots,x_n)}$ refer to?
I have been asking a rather few questions of this nature lately, maybe I'm starting to realise math notation isn't as uniform as I initially thought it would be...
Question: Does this notation
...
4
votes
1answer
173 views
Notation - Two adjacent vectors?
I'm studying multivariable calculus at the moment and have come across equations involving two bolded variables placed side by side, like so:
$$ \nabla \mathbf{f}=\frac{\partial {{f}_{j}}}{\partial ...
1
vote
2answers
274 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 = ...
