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.

Filter by
Sorted by
Tagged with
0
votes
2answers
29 views

Partial derivatives and Galilean transforms

I have trouble understanding the following: \begin{equation} \frac{\partial}{\partial x'} = \frac {\partial x}{\partial x'} \frac{\partial}{\partial x} + \frac{\partial t}{\partial x'} \frac{\partial}...
10
votes
2answers
2k views

How is it possible to differentiate or integrate with respect to discrete time or space?

As far as I have understood, the case is that there is nothing that argues that time or space is continuous, but at the same time we must assume this in order to be able to calculate derivatives or ...
0
votes
2answers
76 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 ...
1
vote
0answers
22 views

Equation for the velocity of a roller coaster at every point

I have been issued a task to create a roller coaster comprised of a piecewise function. In my research I have come across the an equation to calculate the final velocity of the cart found at https://...
4
votes
3answers
2k views

What does “Just before” and “Just after” really mean in physics problems?

So I'm stuck in a dynamics problem that asks what is the acceleration of a body just after A, where A is the point that separates the motion of the body from a curvilinear path to projectile motion. ...
-3
votes
0answers
15 views
-1
votes
2answers
75 views

What does $d$ stand for in this formula?

Context: I am building a tennis ball machine and am having trouble interpreting the following formula for the flight path of the ball. I know all of the other values in the formula but the source I am ...
0
votes
1answer
20 views

If current is charge flow through a point then isn't the surface current always zero?

Current is defined as the rate of flow of charge through a point. Now say we've got a surface charge density $\sigma$ which moves around on a surface. According to the above definition the current ...
1
vote
2answers
51 views

How to find the Taylor expansion of $\vec{r}/r^3$?

I want to show that the Taylor expansion of $\frac{R\vec{e_1}-\vec{y}}{|| R\vec{e_1}-\vec{y} ||^3}$ at $\vec{y}=0$ is equal to $\frac {\vec{e_1}}{R^2}+\frac{3y_1 \vec{e_1}-\vec{y}}{R^3} + O(y^2)$. I ...
3
votes
0answers
83 views

How did the Lagrangian and Hamiltonian theories of motion inspire the idea that forces should be treated as one-forms instead of vectors?

On page-5 of this paper1 by E. Minguzzi titled "A geometrical introduction to screw theory", he writes: Who adopts this point of view argues that it should also be adopted for forces in ...
0
votes
1answer
44 views

How to show functional derivative as a limit of ordinary derivative?

I found this footnote in the appendix (on path integral page 333) of J. Polchinski’s string theory book. can you explain this?
1
vote
1answer
39 views

Problem finding Centre of Mass [closed]

My Question: For finding the Center of Mass ($COM$) of a hollow cone, why do we use its area to define its elemental mass ($dm$) and not its volume, which we use to find the $COM$ of a solid cone. The ...
1
vote
2answers
68 views

Quantum mechanics Dirac delta representation with integral

So I’m doing QM and found bunch of problems for beginners and I’m struggling with this one: $$\lim_{a\rightarrow 0}\int^{\infty}_{-\infty}e^{\frac{ip x}{\hbar}-a x^2}dx=2\pi\hbar\delta(p).$$ If I swap ...
0
votes
1answer
39 views

Can I use the Gauss divergence theorem in a region whose divergence blows to infinity on its surface boundary?

Say we have a vector function $\vec{D}$ defined in some region on whose boundary its divergence goes to infinity and inside we have $\nabla \cdot \vec{D}=\rho$. Then is it valid to use the Gauss ...
1
vote
2answers
47 views

Multivariable chain rule in classical mechanics; example of physical system [closed]

I'm a teaching assistant in calculus and my students who are studying mechanical engineering asked me to explain the multivariable chain rule. So I thought it could be fun if I could give an example ...
1
vote
1answer
30 views

Interpretation of Variation Notes

I would like an explanation to how this Lagragian partial derivative was taken (eq. 3). This probably is more suited for the math Stack Exchange, however this is for a physics course which is why I am ...
0
votes
2answers
71 views

Can anyone suggest a math review book for someone interested in beginning physics study as a hobby?

Good day. This is my first post and I was not sure whether to post here or on Math StackExchange. Since the end product of my goal results in ultimately understanding some basic math in physics, I ...
0
votes
1answer
54 views

Berry Connection Calculation for a 2-Level System [closed]

Suppose we start with a state on the Bloch sphere given by: $$|\psi\rangle = \begin{pmatrix}\cos\left(\frac{\theta}{2}\right)\\e^{i\varphi} \sin\left(\frac{\theta}{2}\right)\end{pmatrix}$$ The Berry ...
2
votes
2answers
39 views

How to create a position-vs-time graph from this?

I'm given the following problem: At what times is the particle found at $x= 20m$? I know this is a very fundamental problem, but still I cannot see how the answer ends up being $4$ and $12$. Note ...
-2
votes
1answer
38 views

Integration used in Derivations

I've seen many derivations in which Integration is used. But I don't understand the fact that why after going to a distance like $y$ or $x$, we take an element $dy$ or $dx$? Instead can't we take any ...
1
vote
1answer
62 views

Alternative formula for the affine connection in a new coordinate basis

In Hobsons's General Relativity: An Introduction for Physicists, pg. 64, he gave two different expressions for the affine connection $\Gamma'^a_{bc}$ in a transformed coordinate basis $x'^a$ (the ...
2
votes
1answer
39 views

How are the SUVAT equations derived? [duplicate]

I hope everyone is doing well and staying safe. So instead of simply memorizing the SUVAT equations, I wanted to find out how the equations are derived to broaden my knowledge. I'm currently a high ...
2
votes
1answer
58 views

Is expectation value of $p^2$ equivalent to this integral?

Let $\psi(x)=Ne^{iax -\frac{m^2x^2}{2} -ibt}$ and I want to compute the possibility of momentum $p$. By definition : $\langle p^2\rangle=\int_{-\infty}^{\infty}\psi^*p^2\psi dx$. Is that equivalent to ...
1
vote
3answers
36 views

Having difficulties deriving the formula for the force acting upon a dam with height $H$ and width $L$

I was recently fiddling around with the derivation of the formula for the force acting upon a dam with height $H$ and width $L$, which in my textbook is derived by integrating the term $dF=p(z)Ldz$ ...
0
votes
1answer
43 views

Acceleration function of position and time

I have an acceleration function in python with position and time parameters and returns the acceleration value. I need the end velocity at a position ,start velocity is zero. how to calculate this ...
23
votes
18answers
5k views

Does it make sense to take an infinitesimal volume of shape other than a cube?

The question clearer: Is the infinitesimal cube the absolute smallest infinitesimal volume? (Sorry if people thought that it meant: "Is it possible and is it done in daily life to use anything ...
0
votes
0answers
45 views

Kinematic formulas for constant $n$th derivative of position

I was wondering how to solve for $x(t)$ in the general case of constant $n$th derivative of $x$. This means to solve the equation $$\frac{\mathrm{d}^n x}{\mathrm{d} t^n}=q,$$ where $q$ is a constant. ...
0
votes
1answer
26 views

Validity of moment of inertia integral in the case of a rod

The wiki says that rotational inertia is defined for point-masses, and by extension continuous bodies. It says: This simple formula generalizes to define moment of inertia for an arbitrarily shaped ...
1
vote
1answer
41 views

Equation for stationary string

I have some doubts on the following derivation of the EOM of a stationary string. Let $F_x, F_y$ be horizontal and vertical tension of the string $\mu$ be the mass per unit length of the string [kg/m] ...
0
votes
4answers
86 views

Acceleration due to gravity during its journey up and down

When we throw an object up into the air, ignoring air resistance, etc, we define acceleration to be -9.8 m/s^2. When it goes down after its journey up, like a parabola, do we define the acceleration ...
0
votes
0answers
33 views

If the integral diverges while finding the expectation value

I was wondering what the expectation value of a wave function might be if the corresponding integral diverges. For example, if I graph the probability density $\Psi (x) = \sqrt{\lambda/\pi} \exp(-\...
0
votes
0answers
28 views

Feynman lectures on physics vol 1: Speed as a derivative (8-3)

I have some trouble understanding what Feynman says here: "The procedure we have just carried out is performed so often in mathematics that for convenience special notations have been assigned ...
2
votes
5answers
179 views

Significance of $\frac{dv}{dx}=0$

Suppose an object is moving with varying acceleration in time. What does it mean when it hits a point where $\frac{dv}{dx}=0$? Does it mean the object has hit maximum velocity? Assume the object ...
0
votes
0answers
21 views

Why is work done in vector fields the integral of the dot product of force and velocity wrt time?

I’ve recently started on a course on engineering calculus and I was wondering why work done is calculated as the integral of the dot product of force vector and derivative of the position vector wrt ...
0
votes
1answer
36 views

How do I calculate the diameter of a liquid poured onto a flat surface?

I'm looking to calculate the diameter of a liquid after being poured on a flat surface, in terms of: Time - t Viscosity - η Density - ρ Volume poured - V Basically, assume you were to carefully pour a ...
1
vote
2answers
64 views

For regular moving objects around us, how many times can I differentiate their position with respect to time until I reach a constant? [duplicate]

When I practise problems, I come across ideal situations like constant velocities, constant accelerations, etc. But in real situations, objects usually don't magically gain momentum or acquire ...
1
vote
1answer
54 views

Calculating the gradient of the dot product of two vectors

I'm trying to calculate $\vec\nabla(\vec k.\vec r)$ where $\vec k =k_x \hat{i}+k_y\hat{j}+k_z\hat{k}$ is a constant vector and $\vec r=x\hat{i}+y\hat{j}+z\hat{k}$ is the position vector. I tried doing ...
0
votes
0answers
24 views

Error Propagations for Integrals and Summations

Suppose I have 3 straight lines: $$y_1 = m_1 x + c_1;$$ $$y_2 = m_2 x + c_2;$$ $$y_3 = m_3 x + c_3;,$$ and $$m_1 = 1 \pm 1.1, c_1 = 1 \pm 1.11,$$ $$m_2 = 2 \pm 2.2, c_2 = 2 \pm 2.22,$$ $$m_3 = 3 \pm 3....
2
votes
1answer
64 views

Boltzmann equation for Photons

I am computing the Boltzmann equation for Photons from the book "Modern Cosmology" by Scott Dodelson. This is the colission term from Compton scattering Then, the Dirac delta is expanded I ...
0
votes
1answer
36 views

How do we calculate the directional derivative of a static vector field? (If there is such a thing.)

So, for a static scalar field $T(x,y,z)$, the derivative along $d\vec l$ is given by $$\frac {dT}{|d\vec l|} = |\vec \nabla T| cos\theta$$where $\theta$ is the angle between $\vec \nabla T$ and $d\...
0
votes
2answers
30 views

Why is easier to measure directly the rate of increase of the volume than the rate of increase of the radius?

I'm reading a Calculus textbook and it says: If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. ...
0
votes
0answers
16 views

Huygen's Principle in Single Slit Diffraction: The Contribution of Each Wavelet

The question I am trying to answer is as follows: "Let $y'$ represent the position of a point within the slit $a$ in Fig. 36.5a, with $y' = 0 $ at the centre of the slit so that the slit extends ...
6
votes
5answers
762 views

Mass-density functions: how is there mass-density at points?

We often discuss mass-density, charge-density, and other such functions: $\rho(x,y)$ for ultra-thin plates and $\rho(x,y,z)$ for 3-d objects. The units for the output of these functions say mass or ...
4
votes
2answers
302 views

Approaching physics using ordinary analysis rather than nonstandard analysis

As far as I know, in physics, calculus is approached using nonstandard analysis in which $dx$, $dy$, etc. (infinitesimals) are treated as fixed, extremely small quantities rather than the standard ...
0
votes
0answers
15 views

Optimum angle for maximum torque in an electric motor taking back emf into account (once it stabilizes)

The standard answer is that maximum torque is when the coil is horizontal, but this assumes a constant current flowing through the coil. When the coil is horizontal, the back emf is a maximum so there ...
0
votes
0answers
22 views

How can I calculate an equation for the final velocity of a spherical ball in a clothoid loop when taking non-conservative forces into account?

I wanted to use the equation to evaluate the final velocities of a spherical ball using coordinates after leaving a clothoid loop section in a roller coaster. Initially, I got this idea from https://...
1
vote
1answer
49 views

Taylor expansion of charge density in Jackson's book

I am learning from Jackson (3r edition), where I found one concept very confusing, that is Taylor expansion of charge density. (This is given in section "1.7 Poisson and Laplace equations" p....
26
votes
5answers
2k views

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 ...
1
vote
6answers
405 views

Why does differentiating a scalar give a vector? [closed]

I was wondering why $F=-\frac{dU}{dr}$ would give me a vector quantity when a scalar quantity is differentiated. There are similar pre-existing queries but I think this issue has yet to be properly ...
1
vote
1answer
69 views

Question about the radial hydrogen eigenfunctions

When calculating the selection rules for electronic transition in the hydrogen atom in dipole approximation, we always focus on the angular integrals. But why the integral $$ \int_{0}^{\infty}[rR_{nl}(...

1
2 3 4 5
12