5
votes
0answers
52 views

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 ...
1
vote
5answers
52 views

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 ...
0
votes
0answers
35 views

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 ...
0
votes
3answers
58 views

Maxwell's Equations Divergence Question

$$ \left\{ \begin{align} \text{div } \textbf{E} & =0, \\ \text{div } \textbf{H} & =0, \\ \text{curl } \textbf{E} & = \frac{-1}{c} \frac{\partial \textbf{H}}{\partial t}, \\ \text{curl } ...
0
votes
1answer
46 views

Maxwell's Equations Curl Question

$$\left\{ \begin{align} \text{div } \textbf{E} &=0, \\ \text{div } \textbf{H} & =0, \\ \text{curl } \textbf{E} & = \dfrac{-1}{c} \dfrac{\partial\textbf{H}}{\partial t}, \\\text{curl } ...
1
vote
3answers
53 views

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 ...
0
votes
3answers
163 views

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 ...
1
vote
2answers
58 views

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.
0
votes
1answer
25 views

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 ...
2
votes
2answers
38 views

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 ...
2
votes
1answer
55 views

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. ...
3
votes
1answer
157 views

Verify that $\nabla(A\cdot B) = (B\cdot\nabla)A + (A\cdot\nabla)B + B\times(\nabla\times A) + A\times(\nabla\times B)$

I'm trying to verify the following identity $$\nabla(\textbf{A}\cdot\textbf{B}) = (\textbf{B}\cdot\nabla)\textbf{A} + (\textbf{A}\cdot\nabla)\textbf{B} + \textbf{B}\times(\nabla\times\textbf{A}) + ...
0
votes
1answer
39 views

Solving diff. equations involving vector product

I've managed to do part (a) by dotting the original equation by $\underline{\dot r}$ however I'm having troubles with part (b): We've learnt very few things to deal with these types of equations ...
1
vote
1answer
33 views

Can you add potentials if charge redistributes?

Let say we have charged conductor $M$ and we know its potential energy function $V_m(r)$ when $M$ is isolated from any charges. We also have charged conductor $N$ with potential energy function ...
0
votes
0answers
85 views

Line Integral Problem best or easier solved using geometry?

Does anyone have any recommendation on a line integral problem involving vector fields (aka work) such that evaluating the resulting line integral using parameterization would be significantly ...
2
votes
1answer
74 views

How would you prove $E = -\vec{\nabla} V$ from the electric potential's line integral?

Basically, we were given an equation: $$V_1 - V_2 = \int_{r_1}^{r_2}\vec{E}\cdot d\vec{r}$$ where $\vec{E}$ is the electric field distribution and $d\vec{r}$ is the displacement vector of the charge. ...
2
votes
2answers
726 views

Integral with cross product

Given a function $\vec{r}(t)$, is there a way to simplify: $$\int_0^u \frac{\vec{r}}{\vec{r} \cdot \vec{r}} \times \frac{d\vec{r}}{dt}~dt$$ With an abuse of notation, it's a line integral like this: ...
6
votes
1answer
190 views

How much can we “cheat” and use vector knowledge in complex analysis?

I'm an engineering-physics student taking a course in complex analysis, and it's a little frustrating, because I see all these connections to vector calculus over the reals (especially as applied to ...
6
votes
1answer
74 views

Solution form for Stokes flows

If $p:\mathbb{R^3} \rightarrow \mathbb{R} $ and $u: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ satisfy: $$\nabla p-\nabla^2u=0$$ $$\nabla\cdot u=0$$ How can we prove that every solution is of the form: ...
2
votes
2answers
80 views

Deriving equations of motion in spherical coordinates

OK, we've been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} ...
4
votes
1answer
171 views

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 ...
0
votes
0answers
101 views

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 ...
1
vote
1answer
154 views

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 ...
1
vote
1answer
307 views

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 ...
0
votes
1answer
49 views

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 ...
3
votes
2answers
245 views

What does δA mean in differentiation?

To be more specific, I met this when doing analytical mechanics involving the principle of least action:
2
votes
2answers
108 views

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 ...
3
votes
2answers
554 views

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 ...
1
vote
1answer
425 views

Electric field of finite sheet: Full analytical solution of integration?

I am trying to work out the integral $$E_{z}(x,y,z)=\alpha\int\int\frac{z\, dx'\, dy'}{((x-x')^{2}+(y-y')^{2}+z{}^{2})^{3/2}}$$ with the limits $$-\frac{a}{2}\leq ...
3
votes
1answer
94 views

Strange integral in 3 space (Maybe Divergence Theorem)

I'm trying to find something called the density of states and the model that I am using specifies $$E = \frac{h^2}{2 m} k^2$$ where $k = |\bf{k}|$. The quantity I am trying to calculate is $$D(E) ...
2
votes
1answer
364 views

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 ...
1
vote
1answer
178 views

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
1
vote
1answer
145 views

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 $B$ ...
3
votes
0answers
77 views

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 ...
4
votes
1answer
157 views

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 ...
4
votes
1answer
225 views

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 ...
3
votes
0answers
128 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
2
votes
0answers
653 views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= ...
2
votes
2answers
77 views

Hint for integral

Could someone provide a hint as to why $$\nabla \cdot \vec a(\vec x) = -i\,\,\,b\,\,\,c(\vec x)$$ where $b$ is a constant, $i$ is $\sqrt {-1}$, implies that $$2\int d^3x \,\,x_ia_j(\vec ...
0
votes
1answer
485 views

Effects of gravity on diffusion [closed]

I'm just now learning the diffusion model and it seems that we aren't taking into account the acceleration due to gravity of the particles. Is this a shortcoming of the model or irrelevant? I don't ...
1
vote
1answer
364 views

Parametric curve of intersection - line integral with respect to arc length

This comes from Apostol's Calculus, Vol. II, Section 10.9 #14: A uniform wire has the shape of that portion of the curve of intersecion of the two surfaces $x^2+y^2=z^2$ and $y^2=x$ connecting the ...
5
votes
2answers
1k views

Line Integral, Work in physics

Hi there all: I have a problem! I need to find the work done on a particle that moves from $(0,0)$ to a point $(1,1)$ by a strait line $y=x$. The force acting upon the particle is $F = (y , 2x$). ...
1
vote
1answer
468 views

del operator - partial derivatives

I'm taking a class in Electromagnetism, and I'm learning about the relationships between voltage and an electric field from Faraday-Maxwell equations. The equation I have trouble with is: $$E = ...
4
votes
2answers
401 views

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 ...
1
vote
1answer
352 views

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 ...
0
votes
1answer
313 views

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 ...
6
votes
2answers
756 views

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 ...
0
votes
1answer
341 views

transforming vector potential with a coordinate rotation

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 ...
1
vote
1answer
122 views

Help with volume integration

I need help solving this integral ($\hat{z}$ denotes the polar axis):$$\int_V\dfrac{\vec{r}\cdot(\vec{r}-c\hat{z})}{|\vec{r}|^3|\vec{r}-c\hat{z}|^3} dV$$ Where $V$ denotes all space. Attempt: $$2\pi ...
2
votes
2answers
422 views

Transforming a sum in an integral

I have a second question about the article "Imperfect Bose Gas with Hard-Sphere Interaction". The authors begins with the sum: $$\frac{{E_2 }}{{E_0 }} = \frac{{16\pi ^2 a^2 \lambda ^2 }}{{V^2 ...