# Tagged Questions

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### Symbolic manipulation inside integral

I'm an undergrad who has just completed the standard calculus sequence (1, 2, and multivariable). I've done well in the courses, however, things like the following, which is a derivation of kinetic ...
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### Vector field ${\bf F}$ with $\int_S {\bf F}\cdot{\bf n}\ dS=c$

Find a vector field ${\bf F}$ on ${\bf R}^3$ with $$\int_S {\bf F}\cdot{\bf n}\ dS=c > 0 \tag{1}$$ where $S$ is any closed surface containing $0$ and ${\bf n}$ is normal Here there is a ...
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### What is the density of a homogeneous disk with mass $m$ and radius $a$?

Could someone help me understand why the density of a homogeneous disk is $\dfrac{m}{(\pi a)^2}$? I am trying to understand an example about finding the moment of inertia of an object. The question ...
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### integral of the sphere describing lambertian reflectance

A Lambertian surface reflects or emits radiation proportional to the cosine of the angle subtended between the exiting angle and the normal to that surface. The integral of surface of the hemisphere ...
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### Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...
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### is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
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### Divergence Computation in Gauge Theories, Knots and Gravity

Hopefully this is just some minor confusion...The first exercise wants us to show that $$\vec \epsilon(t,\vec x)=\vec Ee^{-i(wt-\vec k \cdot\vec x )}$$ satisfies the vacuum Maxwell equations where ...
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### simple question about $\nabla r$

In my physics notes, it says $\nabla r = \underline{e_r} = \frac{\underline{r}}{r}$ and $\nabla \frac{1}{r} = - \frac{\underline{r}}{r^3} = - \frac{1}{r^2} \underline{e_r}$ I don't quite ...
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### Divergence Free Vector Fields that are undefined at the origin

I am aware of vector fields which are undefined at the origin but whose divergence everywhere else is 0. In particular, my students have already seen the inverse square vector field, i.e. ...
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### Surface infinitesimals and its intuitive manipulation?

The excess pressure in the concave side of any liquid bubble or drop with surface tension of the liquid being $T$ is $\frac {4T}r$ and $\frac {2T}r$ respectively. I wanted to derive it using a ...
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### Passing the singularities .

I need some information or detail with example to the following statements. Circumvent the singularity by a contour inside the wave-guide. Circumvent the singularity by a contour outside the ...
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### Direction and Magnitude of a Dog Running

Problem: A dog in an open field runs 12.0m east and then 30.0m in a direction 54 degrees west of north. Part A: In what direction must the dog then run to end up 12.0m south of her original starting ...
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### Direction of the resultant displacement Math help

Problem: A disoriented physics professor drives a distance 3.15km north, then a distance 2.50 km west, and then a distance 1.30km south. Find the direction of the resultant displacement, using the ...
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### Notation question arising from physical problem

I'm a physicist and am a bit confused about the notation in one of the computations here: Let $S = S(U,V,N)$ be a real valued scalar function, and $z = (U,V,N)$. Let $\lambda \in \mathbb{R}$. Given ...
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### What does δA mean in differentiation?

To be more specific, I met this when doing analytical mechanics involving the principle of least action:
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### Travel distance of a particle

How can we show that a projectile fired at an angle $\theta$ with initial speed $v_0$ travels a total distance $\frac{v_0^2}{g}\sin2\theta$ before hitting the ground? The way I set it up is: direction ...
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### Integrating velocity field to get position

I feel silly for simply being brainstuck, but consider the following integral, physically it would be the solution of $\mathbf{p} = \tfrac{d\mathbf{v}}{dt}$ - the position of a given particle in ...
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### Area moment of Inertia and Center of Gravity

Can someone please explain to me once and for all, why is the moment of inertia of a body $A$ Is calculated as: $$I_x = \int_A y^2 dA ,\quad I_y= \int_A x^2 dA .$$ I searched a lot google for a ...
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### Moment of inertia of a circle

A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$ Thank you
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### Let $F(x,y,z) = -c(r/||r||^3)$ be the force resulting from the inverse square law…

$c$ is a constant and $r = (x,y,z)$. Show that $\displaystyle f(x,y,z) = \frac{c}{\sqrt{x^2+y^2+z^2}}$ is a potential function for $F$. What can be concluded from any path from point $A$ to point $B$ ...
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### Using the integral equation, find the eigenvalues and eigenfucntions

The integral equation: $$\int_{-\frac{T}{2}}^{\frac{T}{2}}dt' \phi (t')e^{\Gamma\left | t-t' \right |} =\lambda \phi(t)$$ for $(-\frac{1}{2}T< t < \frac{1}{2}T)$ is useful in photon ...
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### Magnetic fields and the complex plane

The electrostatic potential $\varphi$ must satisfy Laplace's equation in regions without charge: $$\nabla^2 \varphi = 0.$$ If there is no $z$ dependence in the problem we are solving, we can choose ...
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### Solving laplace's equation for an inviscid and incompresible fluid

Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ...
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### What's the relationship between Gauss' law and Newton-Leibniz formula?

Actually it's a puzzle I got in my Physics class. Someone says Gauss' law actually is a specific example of the famous Newton-Leibniz formula, but I couldn't catch it. So far I haven't learned about ...
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### Jacobian matrix normalization

I have a problem with normalization of the Jacobian matrix. There seems to be no clear method for doing it: in some literature, it has been normalized by using some characteristic length, which is ...
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### Help calculating an integral of a vector potential ($\int\frac1rdl$) of a steady current flow through a wire

$\int\frac1rdl$ where $r$ is the position vector from each element $dl$ to the center of the loop with radius $R$. Then I need to take the Curl of that to calculate the magnetic field. I'm having ...
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### Vector physics boat problem

A friend of mine posed this question to me: A boat with a maximum speed of $5$ meters per second crosses a river that is flowing at $3$ meters per second, and is fifteen meters wide. The boat always ...
In electrodynamics, given the vector potential $\vec{A}$, the magnetic field is defined as: $\vec{B} = \nabla \times \vec{A}$ I'm having trouble figuring out how a coordinate transformation (a ...