0
votes
1answer
30 views

Stokes' Theorem and Surfaces

Stokes' Theorem states the following: \begin{equation*} \oint_c \textbf{F}\centerdot d\textbf{r}= \int\int_S (\nabla \times\textbf{F})\centerdot nd \textbf{S}\end{equation*} for a given C that is the ...
2
votes
1answer
58 views

What is the meaning of $d\vec S$ in a surface integral?

Can someone explain if I have a surface $z= 9-x^2-y^2$ What would $\vec{n}$ be? What would $d\vec{S}$ be? Why is $d\vec{S}$ $(2x,2y,1)$ and not $(2x,2y,1)/\sqrt{4x^2+4y^2+1}$? Thanks!
0
votes
0answers
27 views

Area of the portion of the cylinder $x^2+y^2 = 9$ for which $-1 \leq z \leq 2$ and $ 0 \leq \theta \leq \pi/2$

Problem: Find the area of the portion of the cylinder $x^2+y^2 = 9$, for which $-1 \leq z \leq 2$ and $ 0 \leq \theta \leq \pi/2$ I first solved this by parametrizing the surface. $x = 3\cos(u)$ , ...
1
vote
2answers
50 views

Verification of the Stokes theorem for the surface that is a part of a cone

Let $S$ consist of the part of the cone $z=(x^2+y^2)^{1/2}$ for $x^2+y^2\leq9$ and suppose $${\bf A}=(-y,x,-xyz).$$ Verify that Stokes theorem is satisfied for this choice of $\bf A$ and $S$. In ...
3
votes
1answer
48 views

Sketching a surface

If $${\bf F}=2y{\bf i}-z{\bf j}+x^2{\bf k},$$ and $s$ is the surface of the parabolic cylinder $y^2=8x$ in the first octant, bounded by the planes $y=4$ and $z=6$, evaluate $$\int_S{\bf ...
2
votes
1answer
31 views

Parametrizing to Calculate Flux

Evaluate the flux of $\mathbf{f}$ across the oriented surface $\Sigma$ by computing the surface integral $\iint_{\Sigma} \mathbf{f} \cdot d\sigma$, where $\Sigma$ is the surface $z=xe^y$ for $0 \leq x ...
1
vote
2answers
289 views

Boundary under transformation of a closed curve from $R^2\to R^3$

Consider some mapping $\phi: R_{uv} \to S\subset \mathbb{R}^3$ where $R_{uv}\subset \mathbb{R}^2$ and such that it is a simply connected region. We call the boundary of the surface (which we ...
1
vote
0answers
41 views

Difficult Surface Integral

I am trying to perform a surface integral over kind of a weird shape. So the radius of the shape should be equal to the multiple of $3$ constants (one for each of the $x, y$ and $z$ directions) each ...
1
vote
4answers
41 views

Help calculating the surface area given by the polar curve: $r=2(1-\cos\theta)$

I want to calculate the surface area given by the curve: $$ r = 2(1-\cos(\theta)) $$ using an integral. I have thought about doing this: $$ x = r\cos(\theta), \, y = r\sin(\theta) $$ $$ \iint r \,dr ...
1
vote
0answers
14 views

Question about extending a solution to Monge-Ampere solution

I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is: ...
0
votes
1answer
33 views

Understanding surface integrals

My question is a bit vague, but I'm trying to get a better understanding of surface integrals and their relation to physics. Suppose I have a surface, say a sphere, and I have a function which gives ...
1
vote
2answers
27 views

Surface: intersection of 2 polar curves

I have these two polar curves: $$ C_1: r = 2 - \cos(\theta)\\ C_2: r = 3 \cos(\theta) $$ Plots: C1 and C2. I need to find the surface of $D = C_1 \cap C_2$. I started by finding the solution to ...
1
vote
3answers
50 views

surface integral using substitution

I am stuck trying to calculate the following surface integral: $$\int _{R}\int (x+y)^{2}ds$$ over the the following regions: $$0\leqslant x+2y\leqslant 2\: \: \wedge \: \: 0\leqslant x-y\leqslant ...
1
vote
0answers
38 views

Best book for learning multiple integrals, line integrals, greens theorem etc..

I've been searching for a book that teaches multiple integrals and such in a way that I can understand, I need to learn it quickly, so I don't need too much of the intuition, I just need to be able to ...
3
votes
1answer
35 views

Surface area of the part of the sphere $x^2+y^2+z^2=a^2$ that is inside the cylinder $x^2+y^2=ax$

I've been solving some surface area problems lately, but I don't think that the same approach that I was using will work with this one (or at least will result in a lot work). So, I believe I should ...
0
votes
0answers
14 views

Scalar potential, vector field and line integral

I've been trying to get my head around this topic for a project where I need to reconstruct a 3D surface given the estimated normals of it (Photometric Stereo). I just want to be sure I'm ...
0
votes
1answer
22 views

Parametrize plane and get surface area

Find a parametrization of the surface: $y + 2z = 2$ inside the cylinder $x^2 + y^2 = 1$. Then, compute its surface area. I'm having trouble finding the parametrization of the surface. I don't think ...
0
votes
0answers
16 views

How to calculate the double tangential derivative of a surface?

Let z=u(x,y) is smooth surface in $\mathbb{R}^3$. $\vec{\tau}$ is a tangential vector which is orthogonal to the vector of gradient direction $\nabla u$. How to prove the equality $$ u_{\tau\tau} = ...
1
vote
1answer
41 views

Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
2
votes
1answer
42 views

Surface Area Line integral problem

I'm trying to figure out how to solve a surface area with surface and line integrals (showing both methods). The area I'm trying to compute is the area of the shape $$x^2+y^2=9$$ bounded by $z=0$ and ...
0
votes
0answers
45 views

Finding surface integral of the paraboloid and disk

Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle ...
0
votes
2answers
72 views

Surface Area of Two Cylinders Calculus 3

Find the surface area of two cylinders $$y^2 + z^2 = 1$$ and $$x^2 + y^2 = 1$$ I have so far set the two equations to equal $$x= \pm z$$ and $$y= \sqrt{(1-z^2)}$$ I am a little confused on how to set ...
0
votes
1answer
32 views

How to find a parametrization of the set $\left\{(x,y,z): e^x+e^{-x}=z-\sqrt3y, 0<y<x<1\right\}$?

I have to find surface area of set $M=\left\{(x,y,z): e^x+e^{-x}=z-\sqrt3y, 0<y<x<1\right\}$ and my problem is to parametrize it, may you help me?
2
votes
2answers
46 views

How to plot a surface in maple where the range is given by an expression, not constants?

Im trying to plot the surface $z=(1+x^2)/(1+y^2)$ , but specifically the part of the surface that is above $|x|+|y|\leq1$. Cant seem to find any information on how to produce a plot in maple, where ...
3
votes
1answer
41 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
0
votes
1answer
106 views

Find the area of the indicated surface

Find the surface area of the part of the sphere $x^2 + y^2 + z^2 = a^2$ inside the circular cylinder $x^2 + y^2 = ay$ ($r = a\sin(\theta)$ in polar coordinates), with $a > 0$. First time posting ...
0
votes
0answers
41 views

How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
2
votes
1answer
93 views

Flux and Gauss theorem

I have a problem; There seems to be something wrong with my understanding of gauss theorem. Let's say $F = [y ; x^2y; y^2z]$. I want to calculate the flux of $F$ going out of $$D = \{1 \le z \le 2 - ...
0
votes
1answer
27 views

Multivariable Calculus Surface Integral Calculation

I have a surface bounded by $x^2+y^2=1$ and $x^2+y^2=9$ (cylinders) as well as the planes z=0 and z=3.The vector field is $(yx^3,xy^3,x)$. I know this involves the divergence theorem, where I would ...
0
votes
2answers
34 views

Three Surface Integrals

Could someone assist with the following three surface integrals? Q1 The portion of the cone $z=\sqrt{x^2+y^2}$ that lies inside the cylinder $x^2+y^2 =2x$. Q2 The portion of the paraboloid ...
2
votes
2answers
484 views

Surface area of intersection of two cylinders

Let $$R=\{(x,y,z):y^2+z^2\leq 1\,\, \text{and}\,\, x^2+z^2\leq 1\}.$$ Compute the volume of $R$. Compute the area of its boundary $\partial R$. I'm fine with #1. For #2, I have a ...
1
vote
1answer
55 views

Parametrize $|x|+|y|+|z|=1$

How can we parametrize the surface $|x|+|y|+|z|=1$? Here I mean differentiable parametrize. I think we may need to divide it into 8 pieces and consider them respectively.
1
vote
1answer
85 views

bounds for surface integral of a plane

I need to calculate the next surface integral, but i'm having troubles with the bounds; $$\iint -2 \, dS,$$ where $S$ is the part of the plane $x+2y+z=2$ that is cut off in the first octant. My ...
1
vote
2answers
29 views

Does $g(x,y,z)$ (the equation of the surface) need positive $z$ or negative $z$ when doing a surface integral?

$\quad$If a smooth surface $S$ is defined by $g(x,y,z)=0$, then recall that a unit normal is $$\mathbf{n}=\dfrac{1}{\|\nabla g\|}\nabla g,\tag{9}$$ where $\nabla g=\dfrac{\partial g}{\partial ...
0
votes
1answer
36 views

Computing the surface integral of a parabloid

Problem: Solution: I am having difficulty understanding how the author determined the limits of integration of $R$. The author used $\theta=\pi/3\quad to\quad \theta=\pi/2$ and $r=1\quad to\quad ...
1
vote
0answers
111 views

Surface Integral

The glass dome of a futuristic greenhouse is shaped like the surface $z = 8 - 2x^{2} - 2y^{2}$. The greenhouse has a flat dirt floor at $z = 0$ Suppose that the temperature T, at points in and around ...
0
votes
1answer
115 views

Recognize the equation of a surface of revolution

Yesterday, I asked a question about the critic points of the surface $$z = (x^2 + y^2)e^{-(x^2 + y^2)}$$ and my question was if I had a easier way to classify the critic points of this surfaces ...
0
votes
1answer
67 views

Volume of hyperboloid limited by two planes (multivariable calculus).

As usual the teachers solution sheet takes leaps and bounds over steps in the solution that I need to understand it. Q: Determine the volume of the body limited by $x^2+y^2-3z^2=1$, $z=1$ and ...
1
vote
2answers
124 views

Maximum surface area of cylinder (1-variable)

In a given sphere of radius $R$, it is required to find the cylinder with maximum surface area that we can inscribe in this sphere. Using that the radius of the cylinder is $r$, with Pythagoras ...
0
votes
1answer
21 views

Show that the tangent plan pass through the origin

Show that all the tangent plans to the conic surface $z = xf(\frac{y}{x})$ at the point $M(x_o,y_o,z_o)$, where $x_o \neq 0$, pass through the origin of the cordinates First, I've found the tangent ...
0
votes
2answers
454 views

Projection of ellipsoid

Find the projections of the ellipsoid $$ x^2 + y^2 + z^2 -xy -1 = 0$$ on the cordinates plan I have no idea how to do this. I couldn't find much on google to help me with it too. Thanks in ...
2
votes
1answer
145 views

Show that 2 surfaces are tangent in a given point

Show that the surfaces $ \Large\frac{x^2}{a^2} + \Large\frac{y^2}{b^2} = \Large\frac{z^2}{c^2}$ and $ x^2 + y^2+ \left(z - \Large\frac{b^2 + c^2}{c} \right)^2 = \Large\frac{b^2}{c^2} \small(b^2 + ...
1
vote
0answers
101 views

Average arc length between two random points on a unit sphere?

I'm trying to find the average arc length between two random points on a unit sphere. The solution I've come up with is rather ugly. Consider a parametric surface: $$X(u,v)=\sin u\cos v\\Y(u,v)=\cos ...
3
votes
0answers
47 views

Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside a cylinder.

(a) Find the surface integral of $f=|x|-|y|$ over the part of $z=1-\frac{x^2}{M}-\frac{y^2}{N}$ inside the region $\frac{x^2}{M^2}+\frac{y^2}{N^2}=1$ (b) Find the surface integral of $f=|xy|$ over ...
1
vote
0answers
48 views

Gauss and Stocks teory

Given $\phi\in C^1(R)$, and we define the curve and surface $\gamma=${$(x,y):y=\phi(x),0\le x\le 1$} $S=${$(x,y,z):z=\phi(\sqrt{x^2+y^2}),x^2+y^2\le 1$} a.I need to prove that $A(S)=2\pi\int_\gamma ...
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 ...
3
votes
0answers
82 views

Working with projection of areas?

I was recently solving a physics problem which had to do with the momentum imparted by a photon beam to a perfectly absorbing sphere and a perfectly reflecting one. Considering the former and Putting ...
1
vote
3answers
439 views

Find unit normal vector to the surface $z=x^4y+xy^2$ at the point $(1,1,2)$

I've been trying to solve this question: Find a unit vector with positive $z$ component which is normal to the surface $z=x^4y+xy^2$ at the point $(1,1,2)$ on the surface. My working: Let ...
0
votes
2answers
634 views

What is a smooth surface?

What is a smooth surface in terms of tangents and normals? I read in a book that surfaces are smooth if its surface normals depend continuously on the points of that surface. I did not understand this ...
1
vote
0answers
112 views

Finding local maxima of a 2d dynamically created function

So this is a problem I am trying to solve to try and find objects in images. A given image is scanned and the coordinates of detected features are returned. With these coordinates I want to assign a ...