# 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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### Equivalence between Hamiltonian and Lagrangian Mechanics

I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me. The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the ...
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### How to treat differentials and infinitesimals?

In my Calculus class, my math teacher said that differentials such as $dx$ are not numbers, and should not be treated as such. In my physics class, it seems like we treat differentials exactly like ...
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### Rigorous underpinnings of infinitesimals in physics

Just as background, I should say I am a mathematics grad student who is trying to learn some physics. I've been reading "The Theoretical Minimum" by Susskind and Hrabovsky and on page 134, they ...
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### Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?

I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$?
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### How to get distance when acceleration is not constant?

I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question. The equation for distance of an accelerating object with constant ...
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### Infinite series of derivatives of position when starting from rest

Suppose you have an object with zero for the value of all the derivatives of position. In order to get the object moving you would need to increase the value of the velocity from zero to some finite ...
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### Work done in submerging a weightless sphere, I am getting exactly half the correct, can someone point out my mistake!

This is the question: A sphere of radius 0.4 m and having negligible weight is floating in a large freshwater lake. How much work is required to completely submerge the sphere? The density of the ...
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### Electromagnetic field and continuous and differentiable vector fields

We have notions of derivative for a continuous and differentiable vector fields. The operations like curl,divergence etc. have well defined precise notions for these fields. We know electrostatic and ...
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### Landau's derivation of a free particle's kinetic energy- expansion of a function?

I was reading a bit of Landau and Lifshitz's Mechanics the other day and ran into the following part, where the authors are about to derive the kinetic energy of a free particle. They use the fact ...
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### The usage of chain rule in physics

I often see in physics that, we say that we can multiply infinitesimals to use chain rule. For example, $$\frac{dv}{dt} = \frac{dv}{dx} \cdot v(t)$$ But, what bothers me about this is that it raises ...
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### Wrong calculation of work done on a spring, how is it wrong?

So I would have thought that this would be how you derive the work on a spring: basically the same way you do with gravity and other contexts, use $$W=\vec{F}\cdot \vec{x}.$$ If you displace a spring ...
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### What are the differences between the differential and integral forms of (e.g. Maxwell's) equations?

I would like to understand what has to be differential and integral form of the same function, for example the famous equations of James Clerk Maxwell: How to know where to apply each way? Excuse the ...
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### What is divergence?

What is divergence? I was learning about Maxwells equations and don't understand the divergence part of it. Can someone give an intuition of what divergence is in relation to maxwells equation. To ...
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### Newton's original proof of gravitation for non-point-mass objects

Suppose we have two bodies, one very large (Earth), and one very small (a cannon ball). If the cannon ball is some distance away from the Earth, to find out the force produced on the cannot ball, we ...
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### Differentials in Spherical Shell - Maxwell Distribution

In explaining the Maxwell distribution of molecular speeds, my pchem textbook uses the following figure: We are basically trying to find the probability of having a particle with a speed $u$ between ...
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### Derivation of Jefimenko's Equation in Jackson's EMT book

I have been trying to understand the derivation of Jefimenko's equation in Jackson on p.246-247 which can be seen in the photographs attached. First of all I did not fully comprehend the ...
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### Good math books for physicists [duplicate]

In his first lesson (transcripted in "Tips on Physics"), Feynman talks about math for physicists in a very cool and practical way. And at the end of the section he talks something like "so the first ...
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### $\nabla$, $\cdot \nabla$, $\nabla \cdot$, $\nabla^2$ - What do they do? [closed]

I'm trying to teach myself Smoothed Particle Hydrodynamics. Unfortunately, my background is in electronics, so the Navier Stokes equations are somewhat alien to me, as is vector calculus. The video I'...
1 vote
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### I'm having trouble understanding the intuition behind why $a(x) = v\frac{\mathrm{d}v}{\mathrm{d}x}$ [duplicate]

I was shown \begin{align} a(x) &= \frac{\mathrm{d}v}{\mathrm{d}t}\\ &= \frac{\mathrm{d}v}{\mathrm{d}x}\underbrace{\frac{\mathrm{d}x}{\mathrm{d}t}}_{v}\\ &= v\frac{\mathrm{d}v}{\mathrm{d}x} ...
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### Motion with constant speed and constant acceleration magnitude

I was reading this and this posts. From what I gather In 2D: Constant speed $||\dot x||=const$ and constant positive magnitude of the acceleration $||\ddot x|| = const$ imply circular motion. In 3D: ...
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### How does instantaneous velocity cause displacement in just one point? [closed]

I have a question. Falling object graph is curve shape right? And instantaneous velocity is tangent line but how does this velocity make displacement in distance? Because suppose instantaneous ...
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1 vote
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### Area under and slope of the motion graphs

I wanted to ask in general what area under the graph means. Also which physical quantity is highlighted by area under distance vs time graph. I'm confused that area is a 2 dimensional concept and it ...
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### Weird derivative with respect to inverse temperature identity in Tong's statistical physics lecture notes

While reading David Tong's Statistical Physics lecture notes (https://www.damtp.cam.ac.uk/user/tong/statphys.html) I came across this weird identity in page 26 (at the end of the 1.3.4 free energy ...
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### Time derivative of unit velocity vector?

Let's say I have some parametric curve describing the evolution of a particle $\mathbf{r}(t)$. The velocity is $\mathbf{v}(t) = d\mathbf{r}/dt$ of course. I am trying to understand what the expression ...
1 vote
406 views

### How to calculate flux in a helical wire?

We have a wire going around in a helix (just like an inductor) and a constant magnetic field exists along its axis throughout the space. How do we calculate the flux through it? I can't understand ...
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