Questions tagged [calculus]

Calculus is the branch of mathematics which deals with the study of rate of change of quantities. This is usually divided into differential calculus and integral calculus which are concerned with derivatives and integrals respectively. DO NOT USE THIS TAG just because your question makes use of calculus.

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Moment of inertia when a plate is rotated about a perpendicular axis, given other moments of inertia

Consider a rectangular sheet of metal with width $X=2$ m and length $Y=5$ m. The sheet is in the $x\text{-}y$ plane, with the origin right in the geometric middle of the sheet. The $x$-axis is ...
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Taking derivatives of traces over matrix products

I started with evaluating the following derivative with respect to a general element of an $n\times n$ matrix, $$\frac{\partial}{\partial X_{ab}}\left(\mathrm{Tr}{(XX)}\right)$$ I wrote out the ...
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Physical interpretations for the antiderivatives in the solution from D'Alemberts Formula

I am not sure of the physical interpretation for the Fundamental Theorem of calculus as used in: $$ \int_{x}^{y}g(s)ds = G(y)-G(x) $$ where $G$ is the antiderivative of $g$. As an example, apply ...
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Does the logarithm of a non-dimensionless quantity make any sense?

A train consists of an engine and $n$ trucks. It is travelling along a straight horizontal section of track. The mass of the engine and of each truck is $M$. The resistance to motion of the engine and ...
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In the equation: $a = dv/dt$ , is $dt$ the time taken to achieve that instantaneous acceleration?

If you solve for $dt$ from $a = \frac{dv}{dt}$ , is it the time taken to to achieved that instantaneous acceleration? $a$ : acceleration $v$ : velocity $t$ : time
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54 views

When can I treat derivative as a fraction? (Brachistochrone)

My teacher was solving the Brachistochrone problem in class. She parametrized the required path with $x(y)$, then said $T=\int_0^Tdt=\int_{y_1}^{y_2}\frac{dt}{dy}dy=\int_{y_1}^{y_2}\frac{dy}{dy/dt}$. ...
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1answer
79 views

What are some areas of physics, where the concept of “natural integral” may arise?

Natural integral (as we will define it) is a distinguished antiderivative of a function that can be understood as interpolation of the sequence of consecutive derivatives to the $-1$. It has a ...
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34 views

Acceleration vector as a function of Position vector? [closed]

Details given in the question. Initial Position vector was $(x_₀,y_₀)$ Initial Velocity vector $(v_ᵪ,v_ᵧ)$ Magnitude of acceleration - $a$(constant) $x$ is position in horizontal direction $y$ is ...
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46 views

Radioactive decay differential equations [closed]

I am trying to form a differential equation between two different isotopes, Uranium-238 and Thorium-234. The rate of decay of an isotope is proportional to the amount present. So that: $$ \frac{dx}{...
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52 views

Moment of inertia and centre of masses of continuous bodies [duplicate]

I'm quite accustomed with integration and all those calculus involved in finding moment of inertia as well as center of mass. But a random thought is wiggling in my mind. Why do we take $dm$ instead ...
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Can I find the acceleration or velocity when my displacement-time graph is discontinuous?

Today, I encountered the problem where I was asked to find the velocity and acceleration from displacement-time graph but the displacement-time graph was discontinuous. So I am unable to find the ...
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How does a continuous sum turn to an integral? [migrated]

Suppose we have wave packets with different wave numbers. They're in the form of $y=A\cos(kx)$. We can add them together so that $$ y(x)=\sum A_i \cos k_i x $$ for a set of discrete wave numbers. Now ...
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Calculus shouldn't work for describing physics [duplicate]

I am not crazy. Hear me out. I am not from a physics background but from maths. I have a really weird question in physics that is making me lose sleep. How can calculus describe physics? How is it ...
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25 views

Switching the bounds of the potential difference integral

$V = V_C-V_B = - \int_B^C \mathbf{E} \cdot d \mathbf{s}$ My teacher switched the bounds of this integral and put an additional minus sign into it. I couldn't understand because when we change the ...
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20 views

Calculating max percentage error using error propagation?

I was given the actual values of the modulus of elasticity of materials, and I calculated the modulus of elasticity myself using experimental data. I was then asked to "calculate the relative maximum ...
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28 views

$dE$ stands in my way to know the density of states in bulk crystal, how to get rid of it?

In a book about semiconductors, I found the following formula for the density of states: $$D(E)dE=\frac{(2m)^{3/2}E^{1/2}}{2\pi^2\hbar^2}dE. \tag{1}$$ In that book, the important lesson from this ...
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147 views

How does the small angle approximation work for cosine?

In newtonian mechanics equation of motion of a simple pendulum: $$\ddot{\theta}=\frac{g}{l}\sin\theta$$ And then I approximated for small angles $\sin\theta\simeq\theta$ that yields the equation of ...
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1answer
98 views

Chemical potential of ideal gas under gravity [closed]

Here is the question: Consider a monoatomic ideal gas that lives at height $z$ above sea level, so each molecule has potential energy $mgz$ in addition to its kinetic energy. Show that the chemical ...
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1answer
57 views

Thermodynamic equation of differentials (and how to work with them)

Disclaimer: I am not a mathematician, I am a physicist. The thermodynamic identity is usually expressed in the following differential form $$ dU = TdS - PdV + \mu dN, $$ where $U$, $T$, $S$, $P$, $...
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1answer
29 views

Other method for finding the equations of the electric field lines

I have an electric potential which, through separation of variables, can be written as $$\phi (x,y)= X(x) \cdot Y(y) =\sum_{n=0}^\infty Cn\cdot \cos(k_n x)\cdot \sinh (k_n y)$$ with $C_n $ and $k_n$ ...
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3answers
102 views

What type of integrals are these?

Gauss's law $\rightarrow$ $$\oint\vec E\cdot d\vec A=\frac {Q_{encl}}{\epsilon_0}$$ Gauss's law for magnetism$\rightarrow$ $$\oint\vec B\cdot d\vec A=0$$ Faraday's law$\rightarrow$ $$\oint\vec E\...
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2answers
32 views

Creating equation for position when the acceleration depends on the current position

I am writing a simplified simulation of how a drone will move to a target destination. The drone adjusts the acceleration based on the distance from the target location. I want to use the equation ...
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1answer
31 views

Gravitational field intensity of a ring

I am given a ring of a certain radius and mass (assumed to be $R$ and $m$) which is kept in the $yz$-plane with it's axis along the $x$ axis and center at origin. At any point P on the $x$-axis the ...
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1answer
48 views

How to prove the Jackson-Feenberg identity in quantum mechanics?

The Jackson-Feenberg identity for evaluating the expectation of kinetic energy is frequently used in the theory of quantum liquids and is given by: $$<\hat{T}>=-\frac{\hbar^2}{8m}\int d^3r[(\...
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1answer
44 views

Trying to prove that the expression for the radial component of the acceleration is equal to $\mathbf v\cdot \mathbf v/r$

I am trying to prove that the normal component of acceleration of a particle undergoing a curvilinear motion is equal to $\mathbf v\cdot \mathbf v/r$. Here $\mathbf v$ is the velocity of the particle ...
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1answer
75 views

Lorentz Invariance of the Euler-Lagrange equation for fields

Given an Lorentz invariant Lagrangian density $L$ of a Lorentz invariant scalar field $\phi$, How does one show that the following term in the Euler-Lagrange equation is invariant under Lorentz ...
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2answers
62 views

Indefinite integral of a density function

Suppose that $\rho(x)=\frac{dm}{dx}$ is the linear density of a rod. Can we find the mass at each point of the rod by integrating $\rho(x)$, so that:$$m(x)=\int\rho(x)dx.$$ Can we do the same with ...
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1answer
81 views

$\oint{A}=0\implies$ A is a State function?

If $A$ is a thermodynamic variable (ex:Pressure, volume, entropy). then If $\oint{A}=0$, then does it imply that $A$ has to be a state function? I'm trying to prove that Entropy is a state function. ...
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66 views

Is there a textbook for learning physics and multivariable calculus at the same time?

I am a student who took single variable calculus and algebra physics. I want to learn either mechanics or thermodynamics or electromagnetism with multivariable calculus, matrices, lagrange ...
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42 views

Why is work measured linearly with the distance [closed]

If I run and push a car, the force considered is the one felt in my arms not in my legs? The distance the one marked on the field? If the energy outputs is constant and linear with time, why isn't ...
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1answer
68 views

Deriving buoyant force from first principles [duplicate]

How can buoyant force be derived from most basic laws of fluids ? I can think of easy one. Consider such scheme of body floating deep in water : Where $dA$ is elementary downward directed surface ...
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2answers
56 views

How to formalize taking line integral by a reversible or irreversible path?

In thermodynamics work can be done by moving alone a reversible or irreversible path. Physical definitions of reversible and irreversible processes is uncommon in thermodynamics textbooks. The main ...
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1answer
62 views

Vis viva equation - finding how change in velocity changes the semi-major axis distances - math trick [closed]

I'm reading about how to deorbit. I stumped upon a trick here. In the question, we wish to find how a chance in orbital speed changes the semi-major axis distances. So in the answer we start with the ...
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1answer
141 views

Taylor Series of a logarithmic function

I was reading Intro to Modern Statistical Mechanics by David Chandler, on page 63. He states the following: we can expand $\ln\Omega(E-E_v)$ in the Taylor series $$\ln\Omega(E-E_v) = \ln\Omega(E) - ...
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4answers
95 views

I don't understand the logic/concept of $\mathrm dQ=\lambda\,\mathrm dx$. How did we arrive at this expression?

So I've been learning Electrostatics. So while solving for the Electric Field Due to an infinite positively charged rod, I encountered the following expression on the internet wherein the following ...
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2answers
52 views

Changing derivative to difference quotient

Can differential be changed to Delta or difference? In high school education, in the acceleration section of Newton's formula 2, acceleration is a change velocity (velocity difference) divided by a ...
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59 views

This step in Griffiths' Introduction to Quantum Mechanics book

I am working my way through time-dependent perturbation theory at the moment. There is a derivation of the formula for determining the time-dependent coefficients, $c_a(t)$, $c_b(t)$, which I am stuck ...
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50 views

Divergence of vector fields intuitive understanding

I am troubled by the idea of positive and negative divergence of a vector field. I understand that the idea of e.g. positive div is that a gas expands everywhere (for the velocity field of a gas ...
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4answers
1k views

Integrating acceleration - wrong choice of bounds in textbooks?

I've noticed in my physics textbook (and in a lot of other popular sources), that the process of integrating non-constant acceleration to get to a velocity formula, the integrating bounds imposed on ...
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1answer
37 views

Position vs time of electromagnetic force [duplicate]

I don't know if this type of question has been answered and I don't know how to search for it so I am asking it here. Say we have two positive particles in space, with no other forces acting on each ...
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56 views

What does it mean to integrate with respect to matrices?

In Random matrix theory, the following definition of a partition function for an ensemble is common. $$Z=\int dM \exp [-N Tr(M^2)]$$ where $M$ is a Random matrix of dimension $N \times N$. In general,...
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1answer
45 views

Confusion about volume component in Gauss's Law for a cylinder

I am currently working on a problem in which we use Gauss's Law to find the electric field within an infinitely long cylindrical shell (inner radius r, outer radius R) of charge density $\rho = \rho_0 ...
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1answer
84 views

Confusion with work done in isochoric process

So recently I've come up with this thought In an isochoric process, many books state that the work done is zero because there is no change in volume i.e. the gas isn't expanding nor it is contracting....
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1answer
65 views

Derivative of momentum with respect to velocity

Why the derivative of momentum with respect to velocity equals $m^3$? My textbook says: $$d\vec{p}=dp_{x}\cdot dp_{y}\cdot dp_{z}$$ and $$d\vec{u}=du_{x}\cdot du_{y}\cdot du_{z}$$ Then divides the two ...
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1answer
73 views

Lebesgue Integral in physics [duplicate]

I study physics and in this year I have to formule and write my bachelor thesis. I have a lot of ideas but some of them looks more interesting for me. A few days ago I thought about situations in ...
3
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2answers
113 views

Are night and day lengths over a year equal everywhere on earth?

(Question from math.stackexchange, because people told me to better ask here: Original link) As you all know, in winter the nights are longer and there is nearly no sun, but in the summer there are ...
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1answer
126 views

Integration and average in physics? [closed]

Many applications of physics theory involve computations of integrals. Examples are voltage, force due to liquid pressure, surfaces... In some cases, when there is linear dependence between two ...
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1answer
103 views

Higher dimensional version of Stoke's Theorem / Divergence theorem

I've learnt about Stokes' Theorem and the divergence theorem that relate integrals of functions over manifolds to integrals of related functions around the boundary of the manifolds but all in 3-...
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2answers
39 views

Change in areal element

I am reading Griffith's Introduction to Electrodynamics., On example 1.7 while calculating surface integral of $x = 2$ for a cube of side 2., the book states $da = dy \cdot dz$ I don't get this, what ...
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1answer
38 views

Path Coordinates: direction problem (doubt) in derivative of tangential vector

Why is the direction of derivative of tangential vector perpendicular to the direction of the tangential vector?

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