All Questions
208
questions
2
votes
1
answer
45
views
region above the cone and inside the sphere
I've come across a problem where
\begin{array}{c}
\mbox{I have to compute the volume of the region}
\\ \mbox{above the cone}\
z^2 = \sqrt{x^2 + y^2}\ \mbox{and inside the sphere}\ x^2 + y^2 + z^2 = 1
\...
- 75
2
votes
2
answers
76
views
How to apply integration by parts to simplify an integral of a cross product?
I'm reading a physics paper and am trying to figure out how a certain expression is derived (If interested, see Appendix of the paper, Eq. (A7), (A8)). The authors skip a lot of derivation steps and ...
- 23
0
votes
1
answer
121
views
A 4d integral, Exercise I-8 p.68-69 from "Mathematics for the physical sciences" (Dover), Laurent Schwartz
The first part is about two versions of a formula by Feynman for the inverse of a product, one of which being
$$ \frac{1}{a_1\, a_2 \, \cdots\, a_n} = \int_0^1 \int_0^{1-x_1} \cdots \int_0^{1-x_1 -x_2 ...
- 3,679
1
vote
1
answer
27
views
Flux through a paraboloid in the first quadrant
The question is absolutely easy but I am unsure where I go wrong haha. I am given a vector field $\mathbf{u} = (y,z,x)$ and I need to find the ourward flux $\iint\limits_{M} \textbf{u} \cdot d\textbf{...
- 617
2
votes
1
answer
74
views
Integrability of a vector field and its topology
Today, in the lecture, we covered an example of a vector field which suffices the necessary condition for integrability, yet is not integrable. The following field also known as the angular form is an ...
- 617
0
votes
1
answer
34
views
Closed integral curve in vector field implies vector field is not conservative?
I believe, if we have a closed integral curve in a vector field then it is non-conservative. The idea is that say if it were conservative then we have a potential function say $\phi(x)$. Which ...
- 617
0
votes
1
answer
59
views
Deciding a surface in Stokes Theorem
Compute $\int_C (y+z,z+x,x+y) d\vec{r}$, where $C$ is the intersection of the cylinder $x^2 +y^2 = 2y$ and the plane $y = z$.
Is is true that all I can do is apply Stokes Theorem: since $C$ is a ...
- 247
0
votes
1
answer
90
views
Integral of $\nabla\cdot\left(\hat{\mathbf{r}}/r^{2}\right)$
$\nabla\cdot\left(\hat{\mathbf{r}}/r^{2}\right)
=0$ where $r$ is in spherical coordinates and represents the distance from the origin.
In the Griffith' Electrodynamics pg 46, first line it is stated ...
- 121
0
votes
0
answers
46
views
Area of surface intersecting $\vec{f}: \mathbb{R}\longrightarrow \mathbb{R}^3$ and $\vec{g}: \mathbb{R}\longrightarrow \mathbb{R}^2$
Given two parameterized functions of variable $t\in \mathbb{R}$: $$
\begin{cases}
\vec{f}: \mathbb{R}\longrightarrow \mathbb{R}^3 \\
\vec{f}(t)=(x(t),y(t),z(t))
\end{cases}
\quad \quad \begin{cases}
\...
- 480
0
votes
0
answers
38
views
Application of Green's Theorem on a polar closed curve C.
Question: Let $C$ be closed curve as the boundary of region $R$.
$C$ is defined as the polar coordinate inequalities $1\le r\le2, 0\le t\le \pi$.
Define the field $F(x,y)=P(x,y)i+Q(x,y)j$ where $P=x^2+...
- 135
1
vote
1
answer
108
views
Determine the mass of the solid, provided its density at a point $P(x,y,z)$ is proportional to the distance from the $XY$ plane.
I am trying to solve the following problem*
A solid can be described in three-dimensional space by the following system of inequalities:**
$$\begin{cases}
x^2+y^2+z^2\leq1\\
z\geq\sqrt{x^2+y^2}
\end{...
- 1,191
1
vote
1
answer
60
views
Showing that $f(y)=\int_{S^3}{1\over |x-y|^2}|dx|$ on $\Bbb{R}^4$ is harmonic
In a question asking to find a function $f$ such that for $\Vert y\Vert <1$, satisfies $\displaystyle f(y)=\int_{S^3}{1\over |x-y|^2}|dx|$, an included hint suggested finding $\nabla \cdot \nabla f$...
- 2,931
3
votes
1
answer
124
views
Why can the del operator cross product a triple integral be placed inside the triple integral?
Consider the (electric) vector field
$$\pmb{E}(\pmb{r})=k_e\iiint_V \frac{\rho(\pmb{r_s})}{\lVert \pmb{r}-\pmb{r_s} \lVert^2}\frac{\pmb{r}-\pmb{r_s}}{\lVert \pmb{r}-\pmb{r_s} \lVert}d\tau\tag{1}$$
...
- 5,021
2
votes
3
answers
189
views
Electric Field by integral method is not the same as for Gauss's Law
I'm trying to calculate the Electric Field over a thick spherical sphere with charge density $\rho = \frac{k}{r^2}$ for $a < r < b$, where $a$ is the radius of the inner surface and $b$ the ...
- 653
3
votes
1
answer
351
views
Calculating a line integral using Green's theorem in a region with a singularity
Problem:
Calculate the line integral
$$
\int_{A}\frac{y\,dx-(x+1)\,dy}{x^2+y^2+2x+1}
$$
where $A$ is the line $|x|+|y|=4$, travelling clockwise and making one rotation.
Answer: $-2\pi$
Solution:
The ...
- 89