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2
votes
2answers
79 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?
1
vote
2answers
2k 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
0answers
118 views

insulator based gauss law questions

My book is incredibly scarce on insulator based Gauss law questions. Conductors seem to handle themselves pretty simply. Here's a question I'm working on that isn't part of my book. where the radii ...
0
votes
2answers
59 views

Feynman's subscript notation

Consider this vector calculus identity: $$ \mathbf{A} \times \left( \nabla \times \mathbf{B} \right) = \nabla_\mathbf{B} \left( \mathbf{A \cdot B} \right) - \left( \mathbf{A} \cdot \nabla \right) ...
-1
votes
1answer
166 views

Gauss's / Divergence theorem in Classical electrodynamics for the Electric field [duplicate]

Can somebody explain the proof of Gauss's theorem / divergence theorem taking the vector as electric field $$\iiint(\nabla\cdot\vec E)\mbox{ d} V=\iint \vec E \cdot\hat{n} \mbox{ d} ...
-2
votes
2answers
398 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 ...
-2
votes
2answers
119 views

Physical interpretation of $\iiint (∇\cdot\vec E)\mbox{d} V$ [duplicate]

Can anybody explain the physical interpretation of Gauss's law $$\iiint (\nabla\cdot \vec E)~\mbox{d}V~=~\frac{Q}{\epsilon_0}? $$ Also, how is the differential form of Gauss's law obtained from ...
-1
votes
1answer
111 views

Investigation of a pendulum's period, problem creating equation to sum the dynamic velocity

I am investigating the period of a pendulum swing. This is a simple harmonic pendulum and I am already aware of the common, but slightly inaccurate, $2\pi \sqrt{\frac{L}{G}}$ formula. My problem is ...
1
vote
0answers
40 views

Prequisite for the Feynman lectures? [duplicate]

It obviously requires single- and multi-variable calculus and linear algebra, but what else? And where do you suggest to get that background from?this isn't a duplicate because I'm for the math needed ...
3
votes
3answers
605 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). ...
4
votes
4answers
467 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' ...
2
votes
2answers
783 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 ...
19
votes
2answers
903 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 ...
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 ...
4
votes
1answer
114 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 ...
0
votes
2answers
785 views

Electromagnetic wave propagation through two lossless dielectrics

In Elements of Electromagnetics (Sadiku, 3rd edition, Section 10.8), the author says to consider two lossless dielectric materials joined at an interface $z=0$. Here two lossless dielectric materials ...
1
vote
0answers
105 views

Uniform load applied to a parabolic curve

As I have to design the vertebra bone and its natural boundary conditions I came across a problem. How can I applied a uniform load if the place where the force is applied is a parabolic curve. Ok ...
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? ...
7
votes
1answer
481 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 ...
39
votes
3answers
2k views
1
vote
1answer
197 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 ...
2
votes
1answer
132 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 ...
1
vote
0answers
286 views

Apostol or Spivak for mathematical physics? [closed]

I came across many recommendations for both of these books, but I'm not sure which one should I use to study calculus... I know most of the methods used in calculus and I use them frequently, but I'm ...
0
votes
2answers
816 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 ...
0
votes
0answers
120 views

Nicholas Kollerstrom article on the history of Calculus

Today, Newton´s birthday, I read an article posted in the arXiv by Nicholas Kollerstrom http://www.arxiv.org/abs/1212.2666 That basically claims that Newton did not invent Calculus. The article does ...
1
vote
1answer
551 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 ...
0
votes
1answer
478 views

Divergence of cross product of transverse component

If I define the vector as $V_i=V^T_i+V^L_i$ and the transverse part is defined by $$V^T_i=\Big(\delta_{ij}-\frac{\partial_i\partial_j}{\partial^2}\Big)V_j$$ then is is obvious that $\nabla.V^T=0$ as ...
2
votes
3answers
439 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 ...
5
votes
3answers
621 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
3answers
187 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 ...
0
votes
1answer
88 views

Showing $ \textbf{F} \cdot d\textbf{s} = -dV$ is equivalent to $ F_s = -\frac{\partial V}{\partial s}$

Can someone please explain how the following $$ \textbf{F} \cdot d\textbf{s} = -dV$$ is equivalent to $$ F_s = -\frac{\partial V}{\partial s}$$ using some intermediate steps. I don't follow this ...
0
votes
2answers
127 views

Showing $ m\int \frac{d\textbf{v}}{dt} \dot \normalsize \textbf{v}dt = \frac{m}{2}\int \frac{d(v^2)}{dt}{}dt$

Can someone please explain how this equation is valid, using intermediate steps if available? $$ m\int \frac{d\textbf{v}}{dt} \dot \normalsize \textbf{v}dt = \frac{m}{2}\int \frac{d(v^2)}{dt}{}dt$$ ...
1
vote
1answer
150 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
0answers
38 views

Calculus Practice Book for Physicists [duplicate]

Possible Duplicate: Best books for mathematical background? I am a college senior looking to apply to graduate school in hopes of becoming a theoretical physicist. One of the things my ...
1
vote
1answer
240 views

Confused about the physical meaning of velocity variation

I am reading a paper and saw the author wrote something like, Because of no-slip wall assumption, so velocity vector $\vec{v}$ is $0$ on the wall, and also the variation of this velocity on the wall ...
1
vote
1answer
160 views

How to decompose a divergence operator

I am reading a paper, and see someone decompose a divergence operator as follows, could someone judge and see if it is correct? $$\nabla \cdot {\bf{v}} = \left( {{\bf{n}} \cdot \nabla } \right){v_n} ...
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} ...
1
vote
4answers
980 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 ...
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 ...
7
votes
5answers
12k 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 ...