All Questions
171
questions
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
53
views
The surface integral $\iint_S (z^2 + y^2 + x^2) \, dS$ over the cube $S$? [closed]
Evaluate the integral
$$\iint_S (z^2 + y^2 + x^2) \, dS ,$$
where $S$ is the surface of the cube $\{-a < x < a, -a < y< a, -a< z< a\}$.
I've attempted to partition the surface into ...
0
votes
1
answer
64
views
How do I parameterize this double integral?
I am new to multivariable calculus and I am trying to learn how surface area double integrals work. I am stuck on how to parameterize this function:
$\iint_S x dS$ where $S$ is the part of the ...
- 1
0
votes
1
answer
129
views
How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously?
How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously?
This equation is called “Gauss's Law” in physics. I seek for a rigorous mathematical proof for ...
- 174
1
vote
1
answer
63
views
Surface Integral Over Part of Plane
Let $S$ be the surface where $x + 2y + 3z = 0$ and $-1 \leq y \leq 1, 0 \leq z \leq 1$. Compute the surface integral $$\int \int_S (2x,3y-x,1-2y) \cdot \mathbf{\hat{N}}dS$$ where the unit normal ...
- 1,033
3
votes
0
answers
67
views
Struggling with a Calculus III problem :$\iint x+y\ dS,$ where $S(u,v)=2\cos(u)\vec i+2\sin(u)\vec j+v \vec k$ and $0\le u\le \pi/2;\;0 \le v \le9.$
So I have had Calc III many moons ago but I cannot seem to solve this problem for my son who is taking it now. Worked it four ways and got four different answers. Hoping someone here can set me ...
- 31
1
vote
1
answer
107
views
Computing Surface Integral over Tetrahedron
Given a vector field $\mathbf{F}(x,y,z) = (2xz,0,(x-1)^2)$ and three points $P_1 = (1,0,0)$, $P_2 = (0,1,0)$, $P_3 = (0,0,2)$ in $\mathbb{R^3}$. Let $T$ be the tetrahedron with corners in $P_1, P_2, ...
- 1,033
5
votes
1
answer
198
views
Scalar integrals in higher dimensions
The thing I want to do
The typical vector calculus course defines:
A bunch of integrals of vector fields in $\mathbb R^2$ and $\mathbb R^3$: line integrals of a vector field along a curve, flux ...
- 145k
0
votes
0
answers
36
views
Parameterisation of faces of a tetrahedron.
I have to parameterise the faces of the tetrahedron, $z = 0$, $y=0$, $x=y$, $x+z=1$ and use their normal vectors to find the surface integral $\int_Sxy\ dS$. I'm not sure if I have parameterised ...
- 23
2
votes
1
answer
82
views
Is there any way I can calculate this double integral?
$$
S = \int_{-8}^{8} \int_{-\frac{21}{2}}^{21} \left(1 + \frac{1}{8} x^2 + \frac{64}{3969} y^2\right)^{\frac{1}{2}} dy \, dx
$$
I'm trying to finish a high school assignment, and because of the ...
- 21
-1
votes
1
answer
68
views
$\int_D \left(\frac{1}{\Bigr((x+1)^2+(y+1)^2+(z+2)^2 \Bigr)^{10}} - \frac{1}{\Bigr(1^2+1^2+2^2 \Bigr)^{10}} \right) dS$ is positive or negative?
Let $e_1=(1,0,2), e_2:=(0,1,0)$ be the vector in $\mathbb R^3$.
We define $D:=\{(x,y,z)\in\mathbb {R}^3: x+2z=0; x^2+y^2+z^2\leq 1 \}$. I would like to compute exactly the following integral:
$$
A:=\...
- 107
1
vote
1
answer
105
views
Parameterization of one-sheeted hyperboloid to calculate surface area.
Let $S$ be a one-sheeted hyperboloid $S = \{x^2+y^2-z^2=1, z\in(0,1)\}$. Evaluate $\int_S z dA$.
I tried solving this in two methods:
By finding a parameterization: for each $z_0\in (0,1)$, the ...
- 485
4
votes
3
answers
204
views
Computing the integral of a certain surface
I am given the surface $$S:=\{(x,y,z)\in\mathbb{R}^3|\sqrt{x^2+y^2}=2\cosh 2z,z\in[0,1]\}$$
They ask me to find a parametrization for this surface of revolution. This corresponds to the following $$\...
- 657
1
vote
1
answer
102
views
How to calculate the surface area of an object?
Calculate the area of the surface Y given by the equation $z = x^2 + y^2 − 1$ when $z ≤ 0$
Here is my solution:
I got the answer correct but there is something I just did (not toally randomly but I ...
- 1,237
2
votes
1
answer
86
views
Integral over the surface of a paraboloid
The integral I'm trying to solve is the following:
$$ \iint_S xyz \,d\sigma $$ Where $z=x^2+y^2$ and $0<z<1$. So I transform the integral to a double integral: $$\iint_Sxy(x^2+y^2)\sqrt{1+4(x^2+...
- 379