Questions tagged [dimensional-analysis]
The study of the relationships between physical quantities by identifying their units of measure and fundamental dimensions. It is used to convert from one set of units to others such as from miles per hour to meters per second, or from calories per slice of cake to kilocalories per whole cake.
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Can the differential be unitless while the variable have an unit in integration?
Apologies for terminology inconsistencies, as I'm reading a Chinese statistics and probabilities textbook while looking up intrinsics on an English encyclopedia.
This arose when I was reading the ...
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What are the units of a product whose factors themselves contain seperate and distinct units?
Okay so I understand that dividing miles by hours gives us "miles-per-hour", with the logic being that we're splitting up a quantity over a group. But what happens if we keep the same ...
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What is the dimension of this set of points?
The following set of points in $\mathbb{R}^n$ is full-dimensional ($n$-dimensional):
$$\{(x_1,\ldots,x_n)| 0\leq x_i \leq 1 \text{ for all }i\in[n] \}$$
What is the dimension of the following set - is ...
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Solving for a linear equation from a nondimensionalisation
Whilst doing my homework I came across the following question and got particularly stumped at question c). I do not know how I could possibly derive a precise linear relationship. Any help?
When a ...
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Nondimensionalizing and normalizing a partial differential equation
I am trying to understand how to normalize the partial differential equation below. I know how to make it dimensionless but I do not fully understand how to normalize it. I know you're supposed to use ...
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What's wrong with applying our intuition for the behavior of objects in low dimension to high dimension [closed]
The following text is taken from the a book about linear programming that I'm reading:
A graphical illustration is useful for understanding the notions and procedures of linear programming, but as a ...
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Non-Dimensionalizing a Traffic Flow PDE for Physics Informed Neural Network Issue
I'm working on analyzing a traffic flow model described by the following partial differential equation (PDE):
$V_{\max} \left(1 - \frac{2\rho}{\rho_{\max}}\right) \frac{\partial \rho}{\partial x} + \...
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How to "non-dimensionalize" this expression?
As an example, take the following expression:
$\frac{x^2}{k(\frac{x^2}{2k^2}+\frac{5y^2}{4})}-k$
Then:
Define a new variable $K := \frac{ky}{x}$ such that $k = \frac{Kx}{y}$
Plug k into expression ...
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Dimensional analysis of Laplacian
Given a function $$f(x, y): \mathbb{R} \left[kg \right] \times \mathbb{R} \left[K \right] \mapsto \mathbb{R} \left[m \right]$$ where the units of the variables $x, y$ and of the function $f(x, y)$ are ...
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What are the Units of Flux and How do They Relate to Their Physical Meaning
While taking Calculus III, which included some vector calculus, we defined the flux of a vector field $\mathbf{F} \colon \mathbb{R}^3 \to \mathbb{R}^3$ through a surface $S \subset \mathbb{R}^3$ as $$\...
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Calculating coordinates of vertices, given dimensions in an architectural floorplan
So, one of my friend is trying to learn autocad. They were given a floorplan. The floorplan had the dimensions. And they were asked to find the coordinates of the all the vertices of the plan. So we ...
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Dimensional analysis and differential entropy
Differential entropy is a form of entropy that refers to the calculation of continuous distributions. Despite the fact that differential entropy does not have the same properties as the (discrete) ...
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"Axioms" of dimensional analysis
I wondered if there are any theoretical backgound or "formalization" of dimensional analysis. I had an attempt on doing this, by providing some axioms and then "deriving" how ...
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Nondimensionalizing Fourth Order Differential Equation for an Elastic Beam Under Tension
I am going through the textbook A First Look At Perturbation Theory 2nd ed. by James G. Simmonds and James E. Mann Jr.
Exercise 1.14 states: "An elastic beam of section modulus $EI$, resting on ...
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Nondimensionalizing an DE
I am struggling to understand the validity of what is done when you have a differential equation with dimensional variables and you are able to turn it into a differential equation with less ...
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Nondimensionalizing a mixed predator-prey system
I want to Nondimensionalize the following system
$$V'=rV(1-V/K)-aVP,$$ $$P'=-sP+abVP.$$ Which is a predator-prey system where we consider a logistic growth for the prey instead of a malthusian one. I ...
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Suppose that $\Omega\subseteq\mathbb{R}^3$ is a bounded convex region, with the boundary $\partial \Omega$
Suppose that $\Omega\subseteq\mathbb{R}^3$ is a bounded convex region, with the boundary $\partial \Omega$ smooth (i.e. locally $C^\infty$ homeomorphic to the unit disk in $\mathbb{R}^2$). Denote the ...
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The dimension of the unitary matrix as a real variety or a complex variety.
For a complex matrix $O\in \mathbb{C}^{n\times n}$ which satisfies $O^*O=I$, it can be viewed as a real affine variety in $ \mathbb{R}^{2 n^2}$. Let $U$ be the real part of $O$ and $V$ be the image ...
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Try to prove Buckingham's $\pi$ theorem in a simple way
Can I prove Buckingham's $\pi$ theorem like this?
This is the version of the theorem I need to prove:
Consider a model with variables $x_1,...,x_n$, with $k$ fundamental dimensions involved, then $n-k$...
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If there are 29 vehicles per one square kilometers, how many square kilometers per 100 vehicles are there?
29 vehicles/$km^2$ means (1/29) $km^2$/(1 vehicle)
Per 100 vehicles there are 1/2900 $km^2$.
Whatever you sq kilometers are you have to divide by 100 moto vehicles
100/29 $km^2$/100 vehicles
(100/29)/...
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Texts for a mathematical perspective on dimensional analysis.
I am looking for books and papers that take a mathematical perspective on the physics topic of dimensional analysis. I am sure there are such texts out there. I would be very glad to know of any such ...
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Units and $ax^2L^2+bxL+c=0$ in the real world?
It seems that most math equations that come from the real world usually come with dimensions, even though those dimensions are generally ignored. I'm speaking of general dimensions, which include not ...
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Why is it valid to treat units as variables?
I've always taken for granted the fact that units can be treated as variables in mathematical expressions. If you have an object that travels $10m$ in $2s$, you can simply divide the length by the ...
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Can you nondimensionalise a PDE with variable coefficients?
I know how to nondimensionalize a PDE with constant coefficients. For example, if you have a simple diffusion equation on $\Omega \in \mathbb{R}^2$
\begin{equation}
\partial_t u = D \Delta u .
\end{...
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Vector Division in a Basic Physics Problem - Is this defined?
Okay, so I am modelling a problem where I have two objects $a$ and $b$, each with a certain initial position, ${x}_a (0)$ and ${x}_b (0)$, respectively, and each with a certain velocity ${\dot{x}_a}$ ...
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Inequality in theorem proof: Hausdorff dimension and projection theorem with energy integrals (Mattila book)
I am studying Mattila's book "Fourier Analysis and Hausdorff Dimension" and I do not understand how to reach the first inequality in the proof of Theorem 4.2. It is the following:
Let $2<...
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How is this dimensionless parameter obtained?
I have two equations, first says the sum of x and y equals a constant or invariant D,
$$x+y=D$$
Second formula writes the multiplication of those variables x and y in the form of D,
$$xy=(D/2)^2$$
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Is there any more efficient way to find the basis of the intersection of two subspaces?
Let $V = \mathbb R^6.$
Let $W_1$ be the subspace of $V$ spanned by
$$\left ( 1,2,3,4,5,6 \right ) ,\space \left ( 3,4,6,7,9,10 \right ) ,\space \left ( 0,1,0,2,0,3 \right ),\space\left ( 1,-2,3,-4,5,-...
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Theoretical guarantee of binary-relation-preserving embeddings
Suppose we have two sets $A = \{a_1, a_2, \ldots, a_n\}$ and $B = \{b_1, b_2, \ldots, b_m\}$, together with some binary relations between them $R \subseteq A \times B$.
We want to have two embedding ...
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Does the coordinate vectors(Vi)s spans R^n , if S in a set of a n dimensional vector space
Let $S = \{v_1,v_2,\dots ,v_r\}$ be a nonempty set of vectors in an $n$- dimensional vector space $V$.
Prove that if the vectors in $S$ span $V$, then the coordinate vectors $(v_1)S, (v_2)S,\ldots, (...
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