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39
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3answers
2k views
38
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6answers
2k 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 ...
20
votes
2answers
1k 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 ...
11
votes
1answer
570 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 $d=\frac{1}{2}at^2$?
10
votes
4answers
298 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 ...
7
votes
5answers
14k 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 ...
7
votes
1answer
484 views

I reached a result concerning displacement with quantized time intervals. Am I on to something?

A few days ago, I realized a similarity between distance with constant acceleration, $d = v_i t + 1/2 a t^2$, and the sum of integers up to n, $(n^2 + n)/2$. This came up again today when I decided to ...
7
votes
1answer
212 views

Understanding Calculus Notation in Physics

I have just started a first-year calculus-based physics course about electromagnetism and waves. I am having trouble understanding what calculus notation means in the context of physics. Here is a ...
6
votes
4answers
396 views

Name this Mulltivariable Calculus Theorem

In Robert Wald's book General Relativity a multivariable calculus theorem is cited on page 16, which states: If $F:\mathbb{R}^n\mapsto \mathbb{R}$ is $C^{\infty}$ then for each $a=(a^1,...,a^n) \in ...
6
votes
2answers
1k views

Galilean transformation of wave equation

I have this general wave equation: \begin{equation} \dfrac{\partial^2 \psi}{\partial x^2}+\dfrac{\partial^2 \psi}{\partial y^2}-\dfrac{1}{c^2}\dfrac{\partial^2 \psi}{\partial t^2}=0 \end{equation} ...
5
votes
3answers
677 views

Is it safe to ignore derivatives of velocity w.r.t. position and vice versa?

In a certain textbook a function is given as: $$f=f(x(t))$$ And then this is differentiated w.r.t. $t$ to get: $$f_t=\dot{x}f_x$$ (Where the notation $f_u=df/du$, $f_{uu}=d^2f/du^2$, etc.) This ...
4
votes
4answers
490 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' ...
4
votes
3answers
194 views

How empty of fuel are spacecraft booster rockets typically?

A recent XKCD What-if article mentions the situation where each additional kilogram of cargo to LEO requires an additional 1. 3 kilograms of fuel, which in turn requires fuel to carry ...
4
votes
1answer
115 views

Question about Matrix Integral

I am stuck with the technique details of KKN's paper . How to get formula (2.11) $$Z= \int \frac{d \phi }{ \rm{ Vol(G)}} \frac{1}{\rm{Det}(\rm{ad}(\phi)+\epsilon)} $$ from (2.10)? $$Z=\int \frac{d ...
4
votes
1answer
90 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
1answer
98 views

What's the proper interpretation of canceling infinitesimals? [duplicate]

In most textbooks of physics I've found this demonstration of work-kinetic energy theorem: $$\begin{align} W &= \int_{x_{1}}^{x_{2}} F(x)\ dx \tag{1}\\ &= \int_{x_{1}}^{x_{2}} m\cdot a\ dx ...
4
votes
0answers
85 views

On Cohomological Gauge theory Calculation

I am in trouble with calculation details of Witten's Two dimensional Gauge Theories Revisited. My questions is about (3.21) and (3.27). From section 3, we have $$\delta A_i=i\epsilon \psi_i\\ \delta ...
3
votes
2answers
475 views

A basic math identity often used in integrals

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.
3
votes
4answers
2k views

Divergence of $\frac{\hat{r}}{r^2}$

In David J. Griffiths's Introduction to Electrodynamics, the author gave the following problem in an exercise. Sketch the vector function $$ \vec{v} ~=~ \frac{\hat{r}}{r^2}, $$ and compute ...
3
votes
3answers
242 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 ...
3
votes
3answers
667 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). ...
3
votes
1answer
251 views

Three integrals in Peskin's Textbook

Peskin's QFT textbook 1.page 14 $$\int_0 ^\infty \mathrm{d}p\ p \sin px \ e^{-it\sqrt{p^2 +m^2}}$$ when $x^2\gg t^2$, how do I apply the method of stationary phase to get the book's answer. ...
3
votes
2answers
281 views

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 ...
3
votes
4answers
105 views

Based on note, how fast is a stringed instrument's string oscillating?

to be quite honest, I have no idea where to start on this problem mathematically. However, it struck my curiosity and would love to know how it works on a mathematical level. the problem You have a ...
3
votes
1answer
60 views

Boson calculus and Maximum Weight State

I'm just going over a few past exams for tomorrow, and I've come across a question that I'm having quite a bit of difficulty with. Let $\left|0\right\rangle$ denote the Fock vacuum state so that ...
3
votes
1answer
157 views

Neglecting second order differentials

I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials. It's not the first time I have come ...
3
votes
0answers
107 views

Heat transfer from hot water through a copper pipe to oil in a tank

I want to build a geyser attached to a pump with copper tubing running into an oil tank. This is done because the oil solidifies at a temperature below 15 degrees Celsius. I have done a few ...
2
votes
4answers
312 views

Electric Field due to a disk of charge. (Problem in derivation)

This might be a really silly question, but I don't understand it. In finding the electric field due to a thin disk of charge, we use the known result of the field due to a ring of charge and then ...
2
votes
2answers
84 views

Integration of 3-momentum

During a lecture that I missed, I was trapped when the lecturer uses the relation $$dp_x~ dp_y ~dp_z ~=~d^3\mathbf{p} ~=~ 4\pi p^2 dp.$$ Can I know how is this relation derived please?
2
votes
2answers
924 views

Average velocity = Arithmetic Mean

I am having trouble grasping how the average velocity of a particle between an initial position and final position equals to the arithmetic mean of the initial velocity and the final velocity when the ...
2
votes
1answer
109 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, ...
2
votes
1answer
140 views

Partial derivative potential energy of 'free' vibration

I have this rather mathematical question about the calculation of the partial derivative of a potential energy function given by: $$U(x_i)=\frac{1}{2}\sum_{i,j}\frac{\partial^2U(0)}{\partial ...
2
votes
3answers
503 views

How to get an integral formula for the flux time derivative

$$\frac{d}{dt}\int \limits_{A} \mathbf B d \mathbf A = \int \limits_{A} \left( \frac{\partial \mathbf B}{\partial t} + \mathbf v (\nabla \cdot \mathbf B ) + [\nabla \times [\mathbf v \times \mathbf B ...
2
votes
1answer
50 views

First variation of the action in relativistic notation - Landau & Lifshitz “Classical theory of fields”

In Landau & Lifshitz's book, Classical theory of fields, the action for a free particle is defined as: $$\tag{8.1} S= \int ^b _a {-mc \ \text d s}=0,$$ where $$\text d s=c\,\text d ...
2
votes
4answers
90 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 ...
2
votes
1answer
334 views

The curl of a special cross product

When given two vectors $\mathbf{A}$ and $\mathbf{B}$, the curl of the cross product of these two is given by ...
2
votes
1answer
86 views

$\hat{\imath}$ component of force exerted on an electron by a magnetic field?

The magnetic field over a certain range is given by $\vec{B} = B_x\hat{\imath} + B_y\hat{\jmath}$, where $B_x= 4\: \mathrm{T}$ and $B_y= 2\: \mathrm{T}$. An electron moves into the field with a ...
2
votes
2answers
165 views

Infinitesimal volume using differentials

I don't understand why in some texts they put that infinitesimal volume $dV = dx dy dz$. If $ V = V(x,y, z)$ infinitesimal volume should be $$dV = \frac{\partial V}{\partial x} dx +\frac{\partial ...
2
votes
2answers
1k views

Application of Calculus in Physics

Why do we apply Calculus in Physics when most of the quantities are not continuous and are not symmetrical at all levels of magnification? Aren't most, if not all, forms of Matter and Energy discrete? ...
2
votes
1answer
56 views

Taylor series expansion of $\ln$ and $\cosh$ in distance fallen in time $t$ equation

I want to find the Taylor expansion of $y=\frac {V_t^2}{g} \ln(\cosh(\frac{gt}{V_t}))$ I have tried using the fact $\cosh x= \frac {e^x}{2}$ for large t, which works, I just need help on small values ...
2
votes
3answers
211 views

Physical motivation for differentiation under the integral

I am thinking about the mathematical process of "differentiating underneath the integral", i.e. applying the theorem $$\partial_s \int_{-\infty}^\infty f(x,s)\,dx=\int_{-\infty}^\infty \partial_s ...
2
votes
0answers
28 views

Book for multivariable calculus [duplicate]

Hi I want to start learning multi variable calculus specifically for learning electrodynamics. What are some good text books?
1
vote
4answers
1k views

How can/does calculus describe the movement of a particle?

I was talking to Roger Penrose about calculus in the appendix in his book Cycles Of Time and he said I'd need a good understanding of calculus if I wanted to read his book in great depth. He said I ...
1
vote
2answers
3k views

Derive vector gradient in spherical coordinates from first principles

Trying to understand where the $\frac{1}{r sin(\theta)}$ and $1/r$ bits come in the definition of gradient. I've derived the spherical unit vectors but now I don't understand how to transform ...
1
vote
1answer
71 views

Difficulty with the usage of Cauchy's integral formula in Griffiths QM book

On page 410 of Griffiths QM 2nd Ed. book, he begins an analysis to evaluate the integral: $$\frac{1}{2i}\int_{-\infty}^\infty \frac{s \sin{(sr)}}{(s-k)(s+k)}\mathrm{d}s.$$ To exploit Cauchy's formula, ...
1
vote
1answer
198 views

$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf u}$

Please see the next link: http://www3.kis.uni-freiburg.de/~peter/teach/hydro/hydro02.pdf In (2.13), he used: $$\nabla({\bf u}^2)=2({\bf u}\cdot \nabla){\bf u} - 2(\nabla \times {\bf u}) \times {\bf ...
1
vote
2answers
44 views

If the net force on a current loop in a magnetic field is zero, why is torque independent of choice of origin?

Im trying to show that the integral over a closed loop of a crossproduct stays the same if I choose a different origin with $\overrightarrow{r}=\overrightarrow{r}\prime+\overrightarrow{r_0}$ and ...
1
vote
2answers
83 views

Gauss' Law for Magnetism Derivative Form: With or without volume integral?

I've been reading through FLP Vol. II, and he has proven that as the flux through a closed surface is: $\ \int_{surface} \mathbf{F} \space \mathrm{d}\mathbf{a} $, according to the divergence theorem, ...
1
vote
2answers
159 views

Which of these two different forms of spin-orbit interaction is correct?

I am seeing the spin-orbit interaction in two different ways: $\lambda [\mathbf{p} \times \nabla V]\cdot \sigma$ $\lambda [\nabla V \times \mathbf{p}]\cdot \sigma$ I don't see how these two ...
1
vote
1answer
561 views

Derivation of the self gravitational potential energy of a sphere

I have been searching on the Internet but have not found a derivation of the formula for the self gravitational potential energy of a sphere. Can someone show how to do this? I assume it involved 6 ...