0
votes
1answer
17 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
13 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
26 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
33 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
21 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
62 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 ...
2
votes
2answers
34 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
30 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
77 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
32 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
88 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
26 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
1answer
297 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
52 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
58 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
28 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
31 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
69 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
98 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
54 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
101 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
18 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
195 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
118 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
76 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
43 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
45 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 ...
2
votes
0answers
111 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 ...
2
votes
0answers
75 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 ...
0
votes
2answers
312 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
96 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 ...
8
votes
1answer
103 views

Surface integral of $2x+y+2z=16$

Here's the question: Find the surface area of the part of the plane $2x+y+2z=16$ bounded by the surfaces $x=0$, $y=0$ and $x^2+y^2=64$. So, I know I have to parameterize the surface ...
3
votes
0answers
117 views

Calculating the Solid Angle

$\textbf{Problem:}$ Consider we have a class $C^1$ parameterization $\psi:[a_1,b_1]\times[a_2,b_2]\rightarrow\mathbb{R}^3-\{0\}$ for the surface $S$. Also, consider that $S$ is such that the map ...
2
votes
1answer
106 views

tricky surface integral

I am studying for my final and my prof gave us review questions but with no answers so I am lost with this question. If anyone can help I would really appreciate it. Question: Find the area of the ...
0
votes
2answers
898 views

Find surface area that lies above a triangle

Determine the area of the part of the surface $z=2 + 7x + 3y^2$ that lies above the triangle with vertices $(0,0)$, $(0,8)$, and $(14,8)$. I do not know what formula to use to attempt this problem!
0
votes
3answers
204 views

Surface area of a plane inside a cone

Determine the surface area of the part of the plane $z=1+x+2y$ which is inside the cone surface $z=\sqrt{2x^{2}+10y^{2}}$.
1
vote
1answer
116 views

Stokes theorem problem to find alpha and beta so that I is independent of the choice of S

I have a question that I got half through but can't finish it. If anyone could help I would appreciate it. Question: let C1 be the straight line from (-1,0,0) to (1,0,0) and C2 the semi circle ...
0
votes
1answer
171 views

Find area of a curvilinear triangle that includes hyperbolic functions

We were given this question in class and I tried to compute it and it looks to be pretty crazy. Can anyone take a look and let me know if I did it correctly? I would really appreciate it. ...
1
vote
2answers
625 views

Find area of a simple, smooth, closed curve lying in a plane

I was given this question in class and I assume it is a spin off of Green's theorem for finding the area of a closed curve $\lambda$ in 2D but expanded to 3D I believe. Anyways I am pretty confused ...
1
vote
2answers
93 views

How can we parametrize the following surface?

How to parametrize the following surface in $\mathbb{R}^3$: the intersection of $S=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2\leqslant 1\}$ and $D=\{(x,y,z)\in\mathbb{R}^3:x+y=1\}$. Any hints are ...
1
vote
1answer
150 views

Compute the surface area of an oblate paraboloid

Consider the surface S: $z=4-4x^2-y^2, z\geq0$. Compute its surface area. I've tried the following: $Area(S)=\int\int_D \sqrt{(8x)^2+(2y)^2+1}dxdy$ with D being the interior of the ellipse ...
0
votes
0answers
253 views

Area of a surface sphere between two parallel planes

I am given the following question: Consider the surface of a sphere (that is the boundary of the sphere) of radius $R>0$ in $\mathbb{R}^3$ and two parallel planes which are $R$ units away from ...
0
votes
1answer
94 views

Finding The Contour Maps Of A Function Of Two Variables

I am given the function $f(x,y) = \ln|y-x^2|$, and am suppose to find the contour maps. Let $z = c = f(x,y)$. $c = \ln|y-x^2| \rightarrow e^c = e^{\ln|y-x^2|} \rightarrow e^c = |y-x^2|$ I know I ...
2
votes
0answers
140 views

calculating the area on the surface os a sphere created by intersection of two spherical caps!

Consider a spherical object composed of two compartments (A and B, not necessarily hemispheres) sitting at the interface which is characterized by a plane separating 1 and 2. For this case, ...
1
vote
2answers
558 views

What is the difference between surface area and scalar surface integrals?

What is the difference between the surface area of a paremetrized surface and the scalar surface integral of a function in $\mathbb{R}^3$? Are they not the same thing?
0
votes
1answer
103 views

Surface integral

Without getting into the whole question, I was asked to evaluate a surface integral $\iint\limits_S f(x,y,z) da$ where S is the cylinder $x^2 + y^2 = x$ between $z=a$ and $z=b$ Now normally I ...
5
votes
0answers
79 views

surface of a torus by integration [duplicate]

Possible Duplicate: surface area of torus of revolution Let $R>r>0$ fixed. I want to compute the Area of $S=\operatorname{Im} \phi$ given by $$\phi(s,t):= \begin{pmatrix}(R+r\cos s ...
1
vote
3answers
154 views

Would this not take a ridiculously long calculation? (Surface area of parametric surface)

One of the question in my homework asks to verify that the surface are of $ \mathbf{r} = a(1+\cos\phi)\sin\phi \cos\theta \mathbf{i} + a(1+\cos\phi)\sin\phi \sin\theta \mathbf{j} + ...
2
votes
0answers
102 views

How to determine whether or not a specified set is a smooth surface?

I know that a given set $M$ is a smooth surface of dimension $k$ in $\mathbb{R}^n$ iff there's a map $r:U\rightarrow\mathbb{R}^n, U\subset \mathbb{R}^k$ is open such that $\forall a\in U, ...