# 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 ...
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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 ...
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### 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, ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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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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### 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 I(\xi, d) = \int_0^\xi \...
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### 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 ...
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### 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, ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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### 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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