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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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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 ...
Tran Khanh's user avatar
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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} + \...
Proxy's user avatar
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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 ...
Martin.s's user avatar
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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 ...
frogpond The's user avatar
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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 ...
David Gudeman's user avatar
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2 answers
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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 ...
Moyen Medium's user avatar
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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{...
Thede's user avatar
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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 ${\dot{x}_a}$ ...
Adam Gluntz's user avatar
2 votes
1 answer
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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<...
Emilia's user avatar
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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, $$xy=(D/2)^2$$ ...
Luis Porras's user avatar
6 votes
4 answers
147 views

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,-...
mlrofcloud's user avatar
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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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2 votes
1 answer
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Non-zero counts in increasing dimensions

I am working on a presentation that shows the exponential increase as one increases the number of dimensions, and I'm trying to figure out a way to calculate all non-zero or null counts, which I'll ...
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Probability density is dimensional

$P()$ is a probability density function. What does it mean that the density $P(v)$ is a dimensional quantity, having dimensions inverse to the dimensions of $v$? I must be using the wrong definition ...
Hanna Gábor's user avatar
3 votes
2 answers
417 views

Is there a formal mathematical definition of unit systems and dimensional analysis?

I know intuitively what units are and I understand how to DO dimensional analysis fine, but it occurred to me recently that I've never really considered what units or dimensions actually ARE, that is, ...
Mikayla Eckel Cifrese's user avatar
2 votes
1 answer
515 views

Nondimensionalizing a system of PDEs

The following system of PDEs $$\frac{\partial V}{\partial t}+c\frac{\partial V}{\partial x}=\gamma(U-V)$$ $$\frac{\partial U}{\partial t}=\beta(V-U)$$ can be nondimensionalized by change of the ...
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Does Curved Edge exist for a smooth infinitely long right circular Cylinder?

This question is in the continuation of this question. As it is cleared from the comments of the respective question that an infinitely long cylinder which is also a right circular, is a smooth $3$D ...
ankit's user avatar
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1 answer
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Algebraic formula for a time/resource allocation problem?

I initially assumed that this problem could be solved with formulas, but now I'm not so sure. The simply version can be solved easily: Consider a workshop that must complete two jobs. Job A takes 10 ...
Paul Mossman's user avatar
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71 views

Buckingham-$\pi$ doesn't work on Lotka-Volterra

Given the Lotka-Volterra system: $$\frac{dR}{dt}=aR-bRF$$ $$\frac{dF}{dt}=-cF+dRF$$ I think Buckingham-$\pi$ predicts that - since there are 7 variables (R, F, a, b, c, d, t), and 2 units (# of ...
user1143399's user avatar
2 votes
1 answer
664 views

What does it mean to multiply units?

In school we were taught that multiplication is "repeated addition." Of course, that idea breaks down when asked to add 4 to itself -3 times. I have the same intuition that multiplication is ...
Nova's user avatar
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2 votes
1 answer
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nondimensionalize the two mass system

Consider the following two mass system executing rectilinear motion: The first mass, M1, is connected to the left wall by a nonlinear spring with force law: $F_1(x) = −kx − αx^3$ . A linear spring, ...
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How units didn't change while differentiation?

In this example, rate of change has units cm², while the original quantity, area, also has same units. I learnt that units change just like normal ratio, that is dA/dr will have same units as A/r, so ...
Purab Bajaj's user avatar
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2 answers
252 views

Can the areas of a circle and a square be added?

Area of a circle with radius $r$: $A_c = \pi r^2$ has the units $[A_c]=\text{rad}\text{L}^2$. Area of a square with side $a$: $A_s = a^2$ has the units $[A_s]=\text{L}^2$. Since $[A_c] \neq [A_s]$, we ...
ananta's user avatar
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Are 'units' mathematical objects?

I have been studying dimensional analysis and my first question is whether the 'units' i.e the symbols we use are proper mathematical objects, such that the concept of equality makes sense ...
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1 answer
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Checking if an integral converges (or diverges) using dimensional analysis

I have been watching some online lectures in Physics, and the lecturer uses dimensional analysis to make claims such as the following: Consider the integral \begin{equation} I(\xi, d) = \int_0^\xi \...
dsfkgjn's user avatar
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-1 votes
2 answers
327 views

Can you raise a measuring unit to the power of zero?

Say you have a cube with its volume being 27 centimeters cubed. All its dimensions are equal to 3cm, since 3x3x3 = 27 (I know it could have different values, but that isn't the focus here), and we ...
Justquestionasker's user avatar
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13 views

Can we calculate the dimensionality of some discrete space?

Is it possible to calculate the number of dimensions in some discrete space if we have only a complete scheme of all its points and possible transitions between them? There are no regularities, ...
Sergey Miroshnikov's user avatar
2 votes
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Does the Buckingham PI theorem require base units?

When implementing the Buckingham PI theorem, it is common to use basis dimensions such as [MASS, LENGTH, TIME], where each is a base SI unit ...
Reid Johnson's user avatar
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1 answer
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Does the Buckingham Pi theorem depend on rank, or quantity of units?

Does the Buckingham PI theorem require that k be the rank of the dimensional matrix or is it the quantity of base units; given ...
Reid Johnson's user avatar
3 votes
1 answer
99 views

Rigorously distinguishing torque from work, or, a more accurate algebraic structure for dimensional analysis

The algebraic structure underlying dimensional analysis is commonly said to be a finitely generated Abelian group, whose generating set is the set of base units (e.g. length, time, mass, charge, and ...
zwol's user avatar
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1 answer
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Does dimensional analysis require a dependent variable with units?

In a dimensional analysis of a=f(b,c,d), is it possible for a to be unitless? I know that there are limitations on dimensionless ...
Reid Johnson's user avatar
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29 views

Functional Derivatives and their Units

How does taking a functional derivative affect the units of the functional? As an example, considering $\mathcal{F} = \mathcal{F}[h,\nabla h]$ where $\mathcal{F}$ is the free energy functional of some ...
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Can I non-dimensionalize this system of delay differential equations?

I am looking at an application from population dynamics in biology and trying to understand the dynamics of system of delay differential equations below. The model is for a single species with a multi-...
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