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8
votes
5answers
41k views

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 ...
29
votes
2answers
2k views

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 ...
48
votes
7answers
3k views

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 ...
15
votes
3answers
885 views

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 $d=\frac{1}{2}at^2$?
5
votes
5answers
555 views

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 ...
7
votes
6answers
913 views

Is Newton's first law something real or a mathematical formalism?

Why do objects always 'tend' to move in straight lines? How come, everytime I see a curved path that an object takes, I can always say that the object tends to move in a straight line over 'small' ...
5
votes
3answers
940 views

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 ...
2
votes
1answer
255 views

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 ...
1
vote
5answers
1k views

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 ...
1
vote
2answers
2k views

Level of calculus required for physics [closed]

First time for me here so kindly let me know if I violate the rules - especially if this is a duplicate. After reading the page how to become a good theoretical phycist, I started a serious revision ...
10
votes
4answers
614 views

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 ...
4
votes
3answers
2k views

Can someone give an intuitive way of understanding why Gauss's law holds?

Gauss' Law of electrostatics is an amazing law. It is extremely useful (as far as problems framed for it are concerned :D. I do not have a real world-problem solving experience of using Gauss' Law). ...
5
votes
4answers
10k views

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 ...
2
votes
1answer
340 views

Expansion of a function

In Landau-Lifschitz, following expansion is given, We have, $$L(v'^2)~=~L(v^2+2\textbf{v}\cdot\epsilon+\epsilon ^2)$$ expanding this in powers of $\epsilon$ and neglecting powers of higher order, $$L(...
44
votes
3answers
4k views
4
votes
2answers
599 views

A basic math identity often used in integrals [closed]

I'm just wondering about why $y_i=A_{ij}x_j$ implies $$d^Ny=|\det A|d^Nx.$$ I see that $\det A$ is the product of the eigenvalues of a diagonal matrix but still don't exactly see how. Please help.
2
votes
3answers
500 views

Problem in deducing the equations of motion using indefinite integral

As we know, antiderivative or indefinite integral is the function the derivative of which gives the actual function. Let $F(x)$ be the derivative of $f(x)$ ie. the instantaneous rate of change of $f(...
4
votes
1answer
1k views

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 ...
4
votes
3answers
218 views

Is $\dfrac{dx}{dt}$ a fraction or not?

I am new to calculus and during my mathematics class my sir defined $\dfrac{dx}{dt}$ as $$dx/dt=\lim_{t\to t_1}\dfrac{f(t)-f(t_1)}{t-t_1}$$ and my sir made a clear statement that $\dfrac{dx}{dt}$ ...
1
vote
1answer
196 views

Determine the flow and amplitude equation for thermal energy (with Del operator)

It is a question vector calculus and Maxwell's laws. I put it this way. Let's say, we are working in a $3$-Dimensional space ( e.g $x\cdot y\cdot z = 4\cdot3\cdot2$, a certain room/class of that size ...
0
votes
2answers
843 views

Gauss's (Divergence) theorem in Classical Electrodynamics

How does divergence theorem holds good for electric field. How does this hold true- $$\iiint\limits_{\mathcal{V}} (\vec{\nabla}\cdot\vec{E})\ \mbox{d}V=\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \...
0
votes
0answers
36 views

Why does the angular average of the divergence result in a factor 1/3?

In going from eq.(6.13) to (6.14) in http://www.nano.northwestern.edu/intranet/pubdocs/quasiclassical.pdf it is assumed that $$\displaystyle\int\dfrac{d\Omega_{\vec{p}}}{4\pi}\,(\hat{p}\cdot\hat{p})\...