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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64 views

How does the average energy from equation (41.15) of Feynman's lectures approach $kT$ as omega goes to zero?

Feynman said the following equation should approach $kT$ as $\omega$ goes to zero, or $T$ goes to infinity. $$\langle E \rangle = \frac{\hbar \omega}{e^{\hbar \omega/kT}-1}$$ Does anybody know how to ...
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Geodesic equation in terms of four velocity

I am trying to show that for timelike paths, we can write the geodesic equation in terms of the four-velocity $U^\mu=\frac{dx^\mu}{d\tau}$ as $$U^\lambda\nabla_\lambda U^\mu=0.$$ In other words, ...
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Power and work contradiction

A body is starting from rest. A force is acting on it for a short period of time. In that given time, power delivered to it at any instance $t$ is given $$P = F \cdot v_1 = ma \cdot v_1 = mv_1^2/t,$$ ...
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22 views

Particle paths - the distance moved by a particle in a velocity field

This question is is context to particle paths. Particle paths are trajectories of a given particle in the velocity field: $$\boldsymbol{u}(\boldsymbol{x},t)$$ A particle location at position $\...
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Ampere's circuital law for infinitely long wire

I am reading magnetostatics from Introduction to Electrodynamics Textbook by David J. Griffiths So here ampere's circuital law in differential form was derived from biot-sarvart law and an assumption ...
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Taylor Series Expansion of unknown, fraction function

I am learning about deformation, and the deformed state between two points can be defined as $$E(x) = \frac{(f(x+dx) - f(x))^2 - (dx)^2}{2(dx)^2}$$ My textbook says When $dx \to 0$ we can use a ...
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How do I imagine why divergence of curl and curl of gradient is $0$? [migrated]

I tried watching several videos on YouTube, but I failed to gain intuition. I tried to solve it by myself by imagining water flow but I was unsuccessful and got stuck. How do I imagine why divergence ...
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4answers
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Work done in submerging a weightless sphere, I am getting exactly half the correct, can someone point out my mistake! [closed]

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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What's the difference between differentiation and derivation? [migrated]

The question is pretty much straightforward... I just don't get the difference between those two. Is there an easy way of understanding it?
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56 views

Best Calculus one book [duplicate]

I’m currently in my senior year of high-school. I’m planning to major in physics. I really enjoyed basic calculus but I really want to start studying it for real. I know university courses include ...
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Area Swept by particle under central forces is an approximation

From Kepler's second law, we infer, the conservation of angular momentum is equivalent to saying the areal velocity is constant, And the proof goes like this $$ mr^2{\dot\theta=L} $$ where $L$ is ...
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Change of a scalar field/vector field [migrated]

In my book the following is written: The change of a scalar field $du$ in an arbitrary direction, given by an infinitesimal vector with an arbitrary direction $d\vec r$ is calculated: $$du=u(\vec r + ...
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How do I know which equations can be treated as differential equations and which can't?

I'm sometimes mystified by the use of differentials in physics. I don't understand which formulas—on which occasions—can be thought of as differential equations and which cannot. While discussing work ...
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Is the relation “slope=velocity” mathematically valid?

$\text{Slope= tan(angle with respect to positive X-axis)= scalar output}$ $\text{velocity= a vector }$ Source: Hugh D Young_ Roger A Freedman - University Physics with Modern Physics In SI Units (...
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Infinite dimensional generalization of the fundamental theorem of calculus

Can we generalize the fundamental theorem of calculus $$\frac{d}{dx} \int_0^x f(t) dt = f(x)$$ to path integrals $$\frac{d}{dx} \int_{q(0)=0}^{q(1)=x} \mathcal F[q] \, D q = \, ?$$
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1answer
56 views

Is there a difference between instantaneous speed and the magnitude of instantaneous velocity?

Consider a particle that moves around the coordinate grid. After $t$ seconds, it has the position $$ S(t)=(\cos t, \sin t) \quad 0 \leq t \leq \pi/2 \, . $$ The particle traces a quarter arc of ...
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What does $\exp\left( ax\frac{d}{dx} \right)$ do on $\psi(x)$?

I'm trying to find out $$\exp\left(ax\frac{d}{dx}\right)\psi(x)= \ \ ? $$ I tried spending the exponential and then operating the derivatives one by one but I found no pattern. Besides, it gets ...
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Why do I get different results in the draining tank problem?

This is a sketch of the situationWe have to do a small mathematical paper for our school in which we wanted to describe the water that flows out of the cylinder with a differential equation. We also ...
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Toss of a ball using a double integral

Let's say I want to calculate the height of a thrown ball: Let $x''(t)=-g$ and $x(0)=x(T)=0$ and $x'(0)=v_0$. One could then integrate 2 times and it is done. My professor told me to write it this way,...
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2answers
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Finding the centre of mass in polar coords with double integrals

The centre of mass of a body can be found using the general formula: $$ \bar{\boldsymbol{r}} = \frac{1}{M} \int \boldsymbol{r} \ \mathrm{d}M $$ (RHB, p. 195)*. When I try to use this method with polar ...
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1answer
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How to determine the minimum “Arrival Distance” given a maximum velocity, acceleration and jerk along with an initial velocity and acceleration?

Problem Given the following: $A$ - maximum acceleration. $J$ - constant jerk (the rate of change of acceleration). $v$ - initial velocity. $a$ - initial acceleration (where, in practice, $a ∈ [-A, A]$...
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If displacement is 0, does that mean initial velocity equals final velocity?

For instance, one of the kinematic equations is : $$v_f^2 = v_i^2 + 2ad$$ where $v_f$ is final velocity, $v_i$ is initial velocity, $a$ is acceleration, and $d$ is displacement. Say for instance a guy ...
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Force exerted by light on a differential surface

Note: This is not a homework question I simply would like to understand how force is exerted on a differential surface and the question rose up in my mind while solving this problem. Suppose a ...
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2answers
51 views

Derivation of the unit change in resistance

The unit change in resistance is given by this equation but I don't understand how it is derived: $$ {dR \over R} = {d\rho \over \rho} + {2dL \over L} - {dV \over V}. $$ The resistance of a ...
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35 views

How to determine the distance travelled before a maximum acceleration is reached given a constant jerk?

Problem Given: An initial velocity and acceleration of 0. A maximum acceleration $A$ A constant jerk $J$ How might one determine the distance $D$ traversed before the maximum acceleration $A$ is ...
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1answer
40 views

Can a position variable have an infintesimal in it?

I've been pondering unstable systems, such as a perfectly round rock atop a smooth hill. At the top of the hill is a metastable point where the rock could roll either way after an arbitrary amount of ...
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1answer
47 views

Flux of an inverse square field

This question came in my physics test: What is the value of the surface integral $\oint_S\frac{\overrightarrow{r}}{r^3} \,\cdot\mathrm{d}\overrightarrow{A}$ for r>0? The professor says that the ...
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1answer
55 views

How do you express the rate of change with radius of an expression $x$, at a given radius in a gravitational field?

Is it enough to say $dx/dr$, and specify an expression that x refers to, or do you need also to say something like (dx/dr)(r), because it is at a given radius?
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104 views

Is this notation inconsistent? If not, can some explain why not?

Im working through a textbook section on particle kinematics. An example given is relating vertical velocity to horizontal velocity and states: $y$ has a constant velocity of $10 \ \rm [m/s]$ $y=(0....
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Why distance equals initial velocity times time Plus acceleration over two times time Squared? [duplicate]

i am a beginner in physics and I do not understand why is the d=vi(t)+(1/2)a(t^2).
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Problem in instantaneous acceleration and instantaneous velocity

Recently i came accross a problem that said An object is dropped straight down from helicopter the object falls faster and faster but its acceleration decreases over time becoz of air resistance. the ...
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Query regarding instantaneous velocity and instantaneous acceleration

Suppose an object's velocity is $5 \ \text{m/s}$ at $t = 1$ seconds and $8 \ \text{m/sec}$ at $t = 2$ seconds then the acceleration here is $3 \ \text{m/sec$^2$}$ i.e at $t = 1$ seconds the ...
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1answer
30 views

Calculus and trigonometry course

Can anyone please tell me any book or refer any kind of short term course on calculus and trigonometry required for physics.
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Lennard-Jones potential, distance $r$ for minimum energy

I'm sorry if the question seems stupid. I found (wikipedia) that the Lennard-Jones potential has it's minimum at a distance of $$r = 2^{\frac{1}{6}}\sigma.$$ If $U(r)_{min} = -\epsilon$ $$U(r) = 4\...
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73 views

What is the difference between two types of velocity?

What is the difference between $v=\frac{s}{\Delta t}$ and $\bar{v} =\frac{\Delta\bar{x}}{\Delta t}$, are they the same?
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Infinitesimals meaning in this context

I was solving this rocket propulsion's classic mechanics exercise: M is the instantaneous rocket's mass, and v its velocity. The exhaust gases are ejected with speed 𝑢 relative to the speed 𝑣 of the ...
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What is a good book for learning quantum mechanics with mathematical derivations? [duplicate]

I know calculus (ODEs, PDEs, integration over 3 dimensions, limits, W, Zeta functions, series etc) and am working on linear algebra. What it the best book to learn quantum mechanics from with ...
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36 views

Electric field of electric dipole and gradient properties

I am trying to work out whether there is a way to calculate the electric field of a dipole from the following formula: $$\phi(\vec{r}) = -\vec{p} \cdot\vec{\nabla}\phi_0$$ Where $\phi_0$ is the ...
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59 views

Is the lagrangian density convex if the lagrangian is convex?

Let $L = \dot{q}^T M(q) \dot{q} + V(q)$, i.e., the lagrangian has a quadratic form and hence is convex w.r.t to the velocities, considering that $V(q)$ plays the role of a constant. And now let the ...
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1answer
76 views

What does the $d$ mean in metric tensor calculations?

In many metric calculations, like the Schwartzschild metric, we see formulas like $d^2X / dt^2$ and many other formulas with a $d$ in them. You'd be surprised that I've been looking for months to ...
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63 views

Does the SUVAT equations of motion (Kinematics) come from some differential equation?

Wikipedia says about the equations of motion that; "If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.&...
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1answer
28 views

Multivariate analogue of triple product rule

I was doing some thermodynamic calculations and I found the need for a multivariable analogue of the triple product rule. Basically I had a set of $m$ functions $z_i(y, x_1, x_2, \ldots, x_m)$, and I ...
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3answers
227 views

Calculating displacement from acceleration (intuitively) [closed]

If I say acceleration of car is constant at $4\; \rm m/s^2$. Then isn’t it that it covers $4\; \rm m$ in $1\; \rm s$ with velocity $4\; \rm m/s$. Then in $2\; \rm s$, the velocity is $8\; \rm m/s$. ...
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1answer
19 views

Average torque on a Projectile of mass $m$ with initial speed $u$ and angle of projection $θ$ between initial ($P$) and final ($Q$) positions is [closed]

Question is as follows: Average torque on a Projectile of mass $m$ with initial speed $u$ and angle of projection $θ$ between initial $(P)$ and final $(Q)$ positions is I researched a lot but wherever ...
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36 views

Calculate the intensity of light emerging a droplet using calculus

I am reading a paper about rainbow and the appendix here is trying to explain the calculation the author made to calculate the intensity of light. The author mention about it is related to a ...
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3answers
107 views

Confusion in equation of integration and differentiation for motion?

It says displacement time equation here but x is position of particle here I think. It is not displacement of particle here. But when it is given velocity of particle to us and then we integrate it. ...
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2answers
66 views

Why quantities in physics are always talking about rates? [closed]

I get the idea that physics wishes to study changes to discover new rules. But why is everything related to rates? Acceleration,Velocity? Could we use something else apart from these? What can you ...
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1answer
75 views

Finding the form of an Infinitesimal Lorentz matrix

In the context of Lie groups, when looking for the form of the Lorentz generators we expand a general Lorentz matrix using some infinitesimal parameter $\epsilon$ such that $\Lambda = \mathbb{1} + \...
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8answers
6k views

Why do we need instantaneous speed?

I am new to this topic and was just wondering about the use of instantaneous speed. I mean, we use to calculate the speed of car let us say at 5 sec. So we take the distance travelled in 4.9 to 5.0 ...
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2answers
45 views

Calculating for total distance travelled vs position given $v(t)$ graph

It is currently to my understanding that the area under a $v(t)$ graph is the displacement of an object because $$\int v(t)dt = s(t).$$ However, some of the problems I have attempted recently give you ...

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