All Questions
60
questions
0
votes
1
answer
36
views
Parameterising Surfaces Integration
$S$ is the surface of a cube which is bounded by $6$ planes being $x=1,x=3,y=2,y=4,z=0,z=2$. The normal vector points outwards and the vector field is $F = (x^2-sin(yz),\frac{cos(x)}{x^2}-yz,x^2y)$
...
- 261
2
votes
1
answer
398
views
Stokes' theorem on a triangle
I've been given a question that I'm having trouble figuring out:
Calculate
\begin{equation}
\oint_{T} xydx + yzdy + zxdz
\end{equation}
using Stokes' theorem, where $T$ is the triangle with vertices ...
- 143
2
votes
2
answers
156
views
Integration using Stokes' theorem
I have a problem that I can't seem to figure out.
Given a surface $S$:
\begin{cases}
x^2+y^2 \leq 1 \newline
z = y^2
\end{cases}
Let $C$ be the edge of $S$. $C$ is oriented so that the projection of $...
- 143
1
vote
2
answers
120
views
How to calculate the line integral when the vector field is conservative?
Calculate
$$\int_{\gamma} x^3 dx + (x^3 + y^3)dy$$
where $\gamma$ is the line $y=2-x$ from $(0, 2)$ to $(2, 0)$.
It was my understanding that if the vector field was conservative then the path doesn't ...
- 1,237
3
votes
1
answer
97
views
Calculate flow integral using Stokes's theorem
Calculate the flow integral $$\iint_{Y} \text{curl} (\vec{F}) · \hat{N} dS$$
where $Y$ is part of the sphere $x^2 + y^2 + (z − 2)^2 = 8$ that lies
above the $xy$-plane and $\hat{N}$ is the outward ...
- 1,237
2
votes
1
answer
327
views
Verify stokes theorem in spherical coordinates
I got the vector field
$F= r \sin^2 (\theta) \hat{r} + r\sin(\theta)\cos(\phi) \hat{\theta}+ r\cos^2(\theta)\sin(\phi) \hat{\phi}$.
I’m trying to solve the integral
$ \int_C F \cdot dl$ over a close ...
- 25
2
votes
1
answer
101
views
Difficulty with verifying Stokes' theorem for a given domain and a specific differential $(n-1)$-form
Let $f:\Bbb R^{n-1}\to\Bbb R$ be an everywhere positive $C^\infty$ function and let $U$ be a bounded open subset of $\Bbb R^{n-1}.$ Verify directly that Stokes' theorem is true if $D$ is the domain ...
- 4,670
0
votes
1
answer
63
views
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
0
answers
98
views
Integral of a vector field across the intersection of the unit sphere and the plane $Ax+By+Cz=0$ and Stokes theorem
Determine the integral of the vector field $F(x,y,z)=(y,z,x)$ across the intersection of the unit sphere $x^2+y^2+z^2=1$ and the plane $Ax+By+Cz=0$ with an arbitrary orientation.
My attempt:
My idea ...
- 4,670
0
votes
0
answers
48
views
Flux of $\mathbf{f}(x,y,z) = (y-z,z-x,x-y)$
At class, I have been proposed the following exercise:
Consider the vector field $$\mathbf{f}(x,y,z) = (y-z,z-x,x-y).$$ Calculate the flux of $\mathbf{f}$ through the surface $$S = \{x^2+y^2+z^2 = ...
- 65
6
votes
1
answer
300
views
Multivariable Calculus Exam Mistake?
This question was from an exam taken in January 2022 on a course on introductory multivariable calculus and was worded exactly as follows:
"For a general surface $S$ bounded by a closed curve $C$ ...
- 61
1
vote
1
answer
234
views
Calculate Integral of intersection between two surfaces
Calculate
$$\int_{\partial F} 2xy\, dx + x^2 \, dy + (1+x-z) \, dz$$
for the intersection of $z=x^2+y^2$ and $2x+2y+z=7$. Go clockwise with respect to the origin.
My attempt:
I first calculate the ...
- 1,604
0
votes
0
answers
242
views
Stokes' theorem on a paraboloid
I am currently undertaking a class on Classical Electrodynamics, and I am realising my multivariable calculus skills are slightly rusty, as it has been a few years.
Anyhow, I am given a surface
$$S\{(...
- 487
0
votes
1
answer
34
views
Need help computing flux integral of vector field over an unclosed shape
I need help computing this integral:
$$
\iint_{S}
\frac{x \hat{\imath} + y \hat{\jmath} + z \hat{k} }{
( x^2 + y^2 + z^2 )^{3/2} } \cdot \hat{n} \, dS$$
over S that's defined as:
$$S = \...
- 507
1
vote
0
answers
27
views
Help with visualizing the surface to be able to calculate the line integral
We are given the vector field
$F(x,y,z)=[2z+y, 2x+z, 2ycos(z)]$
A surface S is composed of two parts. One part $S_1$ given by $z=x^2+y^2$, for $0\leq z\leq4$ and $0 \leq y$. The second part $S_2$ is ...
- 547