# 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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### Converting differential to gradient

Landau & Lifschitz's fluid mechanics book proposes the following statement for an isentropic proccess: $$dH=vdp \Rightarrow \nabla H=v\nabla p$$ What's the rigorous way to get this result (...
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### Leibniz's Theorem [closed]

I'm not familiar with Leibniz's Theorem, and by the time I added my substitutions, I got lost in the variables and how they are suppose to transform. Please help?
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### Thermodynamic adiabatic process, question regarding mathematical operations

I have a question regarding mathematical operations often seen in physics books: In an adiabatic process the heat is 0, so by the first law of thermodynamics we have that $E = W$, and an infinitesimal ...
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### Circular motion equivalent in three dimensions [closed]

Are there equations or even a concept of circular motion/tangential acceleration/centripetal acceleration in three dimensions? Maybe something called "spherical acceleration"? or am I just ...
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### Prove that Gauss's Law Holds Under Translations

We know that Gauss's Law says $\nabla \cdot E(\textbf{x}) = \frac{\rho(\textbf{x})}{\varepsilon}$, and we also know that this relationship should be true regardless of where you're located in three ...
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### Derivative of distance [duplicate]

I know that $speed = |\frac{\vec{dr}}{dt}|$ and first derivative of distance with respect time will be $\frac{d\vec{|r|}}{dt}|$ These 2 expressions don't seem to represent the same thing. But when I ...
1 vote
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### Limit definition of scalar curvature for flat vs curved space in 2D, 3D and so on in Zee

In Zee's book, Einstein Gravity in a Nutshell, p. 6 + p. 77, he says that \begin{equation} R = \text{lim}_{\text{radius} \rightarrow 0} \frac{6}{(\text{radius})^2} \left(1 - \frac{\text{...
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### Finding the distance under a position-time graph

I have to find the distance travelled by a particle between t=0 and t=4 secs for x(t)= 4t^2 - 2t^3. I tried differentiating the equation to find v(t) then integrating it within t=0 and t=4, but that ...
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### Is $n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} =\frac{60}{2\pi}\sqrt{g\frac{\int y_idx}{\int y_i^2dx}}\quad ?$

I have a question about this formula used to calculate the first critical speed of a drive shaft. $$n_{cr}=\frac{60}{2\pi}\sqrt{g\frac{\Sigma m_iy_i}{\Sigma m_iy_i^2}} \tag {1} \quad .$$ It is the ...
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### Solving an inexact differential

So we have something called an inexact differential which is when a function is path dependent meaning I can't just subtract the initial and final states to get an answer. Take for example work which ...
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### Deriving Work-Kinetic Energy Theorem

I am currently reading Physics for Scientists and Engineers (Ninth Edition) by Serway and Jewett and in Chapter 7.5, a derivation of the work-kinetic energy theorem was shown. To give context, ...
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### Advection term for a matrix equation

How can I calculate a quantity like $(\vec{v} \cdot \nabla) M$ where $\vec{v}$ is the velocity vector, and $M$ is some 3x3 matrix? (if one wants, assume $M$ is a tensor) This would be the advective ...
44 views

### Determine the meaning of a gradient of a graph [closed]

How do you determine the gradient of a graph in physics, such as how with a velocity-time graph, the gradient is acceleration. I want to know the general method for figuring out what the differential ...
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### Taking the second time derivative of a scalar field

Given some scalar field $\phi(x,y,x,t)$, taking its first total derivative we get: \frac{d\phi}{dt}=\frac{\partial\phi}{\partial t}+\frac{\partial\phi}{\partial x}\frac{dx}{dt}+\frac{\partial\phi}{\...
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### Why can we change $dt$ with $(dt/dp)_s dp$?
In my homework assignment there's the following question: A general thermodynamic system is being compressed isentropically from pressure $P_i$ to $P_f$ while keeping the number of particles constant....