All Questions
63
questions
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Evaluating $\displaystyle\iint\limits_{A}\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)\cdot\overrightarrow{n}~\mathrm{d}S$, where $A$ is the unit sphere
This is from UCHICAGO (GRE Math Subject Test Preparation), Week $5$, Problem $14$.
Let $A$ be the unit $2$-sphere in $\mathbb{R}^3$. Let $\overrightarrow{F}=\left(x^3-y^2z^4,2y^3,z^3-3y^2z\right)$ be ...
- 5,450
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
1
vote
1
answer
69
views
Calculating Electric Flux Through a Closed Surface
I'm trying to solve a problem involving the calculation of electric flux through a closed surface, but it's my first time attempting such a problem and I could use some guidance.
Any help would be ...
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
95
views
Integral Change of Variables: g(x) appears twice
I would like to simplify this integral, and I think I can perform a change of variables. Here's the integral. $t, y$ and $z$ are all scalars:
$$
\int_{0}^{1}f(ty,tz)\cdot z dt
$$
Note that we have a &...
- 280
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
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votes
1
answer
63
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Calculate $\int_{\partial V} \mathbf{F}\cdot d\mathbf{S}$ where $\partial V$ is the border of the union of two sets and $F$ is a flux
Consider the sets $$V_1 = \{ (x,y,z) : x^2 + y^2 + (z-1)^2 \leq 4 \} \ \& \ V_2 = \{ (x,y,z): x^2 + y^2 + (z + 1)^2 \leq 4 \}$$ Let $V = V_1 \cup V_2$, and the vector field $F(x,y,z) = (x + xy, yz,...
- 402
1
vote
1
answer
42
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calculating force on a non-conservative vector field
I have this question as a calculus assignment and have been stucked on solving it for some time. so this is the question:
let F be a vector field:
$$F(x,y,z)=((2/π)xsin πy) i + (x^2 cos πy + 2ye^{-z})...
- 176
0
votes
1
answer
74
views
How to calculate the integral of $\vec{a}\cdot\nabla(\nabla\cdot\vec{a})$?
I want to calculate the following integral:
$$S = \int\vec{a}\cdot\nabla(\nabla\cdot\vec{a})\,\text{d}V$$
I tried to calculate the integral of the $i$-th term, i. e., $\int a_i\partial_i(\nabla\cdot\...
- 13
6
votes
1
answer
300
views
Multivariable Calculus Exam Mistake?
This question was from an exam taken in January 2022 on a course on introductory multivariable calculus and was worded exactly as follows:
"For a general surface $S$ bounded by a closed curve $C$ ...
- 61
2
votes
1
answer
272
views
Why isn't this closed loop curve integral $0$?
I had to solve a problem where I had to calculate the work done by the force field given by:
$$ \vec{F} = \frac{(-y,x)}{x^2+4y^2}, (x,y) \neq (0,0)$$
where we travel along the whole unit circle in a ...
- 1,847
5
votes
2
answers
185
views
How to find this double integral?
Given $\vec F=y\hat i+(x-2xz)\hat j-xy\hat k$ evaluate
$$\iint_R(\nabla \times\vec F)\cdot \vec n dS $$
Where $S$ is surface represented by $x^2+y^2+z^2=a^2$ for $z\ge 0$
My attempt:
i found curl
$$\...
- 1,256
0
votes
0
answers
55
views
Find the potential of a given vector field at a point - getting expressions I can't compute.
the given vector field $$\vec{F}(x,y) = \frac{(x-y)\hat{i} + (x+y)\hat{j}}{x^2 +y^2}$$
has a potential function $U(x,y)$ that is defined in $D=\{(x,y): 1\leq x^2 + y^2 \leq16,\: y\geq 0\}$. and it's ...
- 507
3
votes
4
answers
319
views
What does the notation for a line integral of a vector field actually mean?
I have been told that the line integral of a vector field, F(r) along a curve $C$ is:
$$I =\int_C\textbf{F}\cdot \text{d}\textbf{r}=\int_C(F_x,F_y)\cdot (\text{d}x,\text{d}y),$$
where $\text{d}\textbf{...
- 138
4
votes
2
answers
6k
views
How do I check if the normal vector is pointing inside or outside?
Lets say I've a sphere $x^2+y^2+z^2=1$ and I need to solve an integral $\iint_S\vec F\cdot\vec ndS$. while $S$ is the sphere.
And I can't use gauss law because $\vec F$ is not continuous at some ...
- 2,113