Questions tagged [cylindrical-coordinates]
Questions on cylindrical coordinates, a coordinate system where points in space are represented by their distance to the $z$-axis ($r$), the angle the line joining the orthogonal projection of the point on the $xy$-plane and the origin makes with the positive $x$-axis ($\theta$) and the $z$ coordinate of the point.
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14 views
velocity gradient in cylindrical coordinate.
In cartesian co-ordinate ($x,y,z$), gradient of velocity($\mathbf{u}=(u_x,u_y,u_z)$)(Jacobian matrix)is defined:
\begin{equation*}
\nabla \mathbf{u}=
\begin{pmatrix}
\partial_x u_x && \...
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1answer
30 views
Find $\iiint_V z$ with $V=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$
Let $f(x,y,z)=z$ and $T=\lbrace(x,y,z) \in \mathbb{R^3} : y\geq0, z\geq0, x^2+y^2+z^2\leq 2, x^2+y^2\leq1\rbrace$
Find $\iiint_T f(x,y,z) dV$
I'm having a few problems with this integral, here's ...
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0answers
20 views
Convert cylindrical coordinate displacement to cartesian
I have a set of points of a finite element mesh which when inputted into a solver (ansys) gives the displacement of each node. I can get the displacement values of each node either in r,theta(in rads),...
2
votes
1answer
43 views
Finding the volume of the intersection of a cylinder and a sphere
Let $\alpha>0$. Given the cylinder
$$B_1=\{(x,y,z)\in\mathbb R^3\;|\;x^2+y^2\leq\alpha x\}$$
and the sphere
$$B_2=\{(x,y,z)\in\mathbb R^3\;|\;x^2+y^2+z^2\leq\alpha^2\},$$
how can one find the ...
1
vote
1answer
33 views
Volume integral over circle not around the origin
For an assignment I am asked to find the volume of a given volume R, namely
$R=\left\{(x,y,z):0\leq z\leq\sqrt{4-x^2-y^2},(x-1)^2+y^2\leq1\right\}$.
I have attempted solving this using cylindrical ...
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0answers
15 views
How do I graph using cylindrical coordinates?
I need to graph the curves $x^2 + y^2 = 4$, and $x^2 + y^2 = 25$ in the cylindrical coordinate system, but I don't know how. I substituted $x$ and $y$ values for their cylindrical counterparts, but I ...
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1answer
15 views
Find the volume of the Region (Triple Integral with Cylindrical Coordinates)
Question: Find the volume of the region that is contained by the cylinder $x^2+y^2=81$, bounded above by $z=x$ and below by the $xy$-plane.
I have tried the integral $\int_{0}^{2pi}\int_{0}^{9}\int_{...
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2answers
50 views
Triple integration for the volume of a given sphere
I have a problem which I've had a look on "Maths Stack Exchange" and other resources to help, but still am stuck, so any help would be most appreciated.
My Problem:
Set up a triple integral for the ...
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3answers
76 views
What is the volume above the cone $z= \sqrt{x^2+y^2}$ and bounded by the spheres $𝑥^2+y^2+𝑧^2=1$ and $𝑥^2+y^2+𝑧^2=4$?
What is the volume above the cone $z= \sqrt{x^2+y^2}$ and bounded by the spheres $𝑥^2+y^2+𝑧^2=1$ and $𝑥^2+y^2+𝑧^2=4$?
I tried converting each equation to cylindrical coordinates: $z= $r, $r^2+𝑧^...
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0answers
22 views
Find the volume of solid region bounded by three cylinders.
The equations of the 3 cylinders are given by $x^2+y^2=1$, $y^2+z^2=1$ and $x^2+z^2=1$. While it is common to solve it in cylindrical coordinates via triple integral, I would like to know how to ...
1
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1answer
27 views
Center of mass of a planar lamina
I have to find the center of mass of a planar lamina bounded by $x=0$, $y=1/2$, and $y=x$, with the density of the lamina being $x/(1-y^2)^{1/2}$.
I ended up drawing a picture, and it looks like a ...
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1answer
36 views
Triple integral conversion to cylindrical coordinates equals zero
I'm asked to convert this integral:
$$\int^1_0\int_{-\sqrt{1-y^2}}^0\int^{2-x^2-y^2}_0 x\ dz\ dx\ dy $$
to cylindrical coordinates. This is what I calculated for the limits:
$$\int^{2\pi}_0\int^0_{-1}...
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2answers
47 views
What's The Cylindrical coordinates for this $\int_0^6\int_{-\sqrt{6x-x^2}}^{\sqrt{6x-x^2}}\int_0^{6x-x^2-y^2}\left(x^2+y^2\right)dzdydx$
I want to convert this to cylindrical coordinates
$$V=\int_0^6\int_{-\sqrt{6x-x^2}}^{\sqrt{6x-x^2}}\int_0^{6x-x^2-y^2}\left(x^2+y^2\right)dzdydx = 486π$$
I want to write it like this:
$$V=\int_{\ }^{ ...
3
votes
1answer
62 views
Divergence of a vector field in cylindrical coordinates
Let $\bar{F}:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a vector field such that $\bar{F}(x,y,z)=(x,y,z)$. Then we know that:
$$\nabla\cdot\bar{F}=\frac{\partial\bar{F}_x}{\partial x}+\frac{\partial\bar{...
1
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1answer
71 views
Lagrangian mechanics- conservation of energy
Consider a single particle system whose Lagrangian remains the same if the position of the particle is simultaneously (i) rotated by an arbitrary angle s about the z-axis and (ii) shifted by an amount ...
1
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1answer
63 views
Equations of Motion in Cylindrical Co-ordinates
I've run into an interesting set of differential equations, that I'm not 100% sure where to begin- I'm not looking for a 100% complete solution, more just a push in the right direction of where I can ...
2
votes
0answers
84 views
Laplace transform of heat conduction PDE in cylindrical coordinates.
I'm trying apply the Laplace transformation to solve the non-dimensional heat conduction PDE for a hollow cylinder with convection boundary conditions and a non-homogenous initial condition.
$$\frac{...
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0answers
33 views
Find radius of best fit cylinder from cloud
I want to find the radius of the cylinder that best fits a cloud of points.
I used 3DReshaper to calculate the radius of the best fit cylinder for this points (format: x y z):
...
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0answers
41 views
lateral surface area of cylinder
Use cylindrical coordinates and multivariable calculus to prove that the lateral surface area of a right, circular cylinder with radius 2 and height h is 4pih.
I parameterized x = rcostheta, y = ...
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1answer
13 views
deducing an area of integration in cylindrical coordinates
If I am given an area characterised by $0 \leq z \leq 1-r^{2}$ from this how can I deduce the radius r and the angle with the x-axis, $\theta$ that will span the are and I can ten integrate over i.e. ...
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1answer
49 views
Heat equation in cylindrical coordinates at origin
I'm trying to solve a heat equation in cylindrical coordinates
$$\dfrac{\partial u}{\partial t} = a \left(\dfrac{\partial^2 u}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial u}{\partial r} + \dfrac{1}{...
1
vote
1answer
36 views
Integration with substitution to cylindrical coordinate
Solve the integral
$$
\\A:= \int_{{x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}\leq 1}{x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}dxdydz \
$$
In my solution I have substituted to cylindrical ...
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votes
1answer
46 views
Change of coordinates vs change of shape
It is an elementary fact that we are able to change coordinates system to new one. for example in Cartesian coordinates $x^2+y^2=1$ illustrates a circle. Changing to polar coordinates, this equation ...
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1answer
24 views
how to find the volume of the Revolved Domain about z Axis [ volumes ] [ integrals ]
let $D$ = {$(x,0,z) | (x-1)^2 + z^2 \leq 1$} find the volume of the body obtained by revolving $D$ about the $ Z $ axis.
how do i solve this with integrals ( triple / double ) .
intuitive solution ( ...
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0answers
42 views
Transformation from global spherical to local cylindrical
I have the coordinates of a vortex in a global spherical coordinate on the surface of a sphere defined in terms of latitude and longitude(earth centric). Now I need to transform to a local coordinate ...
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2answers
44 views
Find the volume of intersection between cylinders
Find the volume of intersection of the cylinder
{$ x^2 +
y^2 \leq 1 $} , {$ x^2 + z^2 \leq 1$}, {$ y^2 + z^2 \leq 1$}.
i am having tough time finding the volume how do i solve this kind of ...
0
votes
1answer
30 views
Transformation from spherical to cylindrical coordinates
I have a coordinate on the surface of a sphere i.e latitude and longitude(two dimensions). I need to transform into a 2-D cylindrical coordinate frame i.e. r and the azimuth angle theta. Is there a ...
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0answers
21 views
With $\alpha>0$, how to calculate the volume of $E_\alpha=\{(x,y,z)\in\Bbb R^3: x^2+y^2-\alpha^2\le\alpha z\le\alpha(3\alpha-2x)\}$
With $\alpha>0$ let $E_\alpha=\left\{(x,y,z)\in\Bbb R^3:\frac1\alpha(x^2+y^2-\alpha^2)\le z\le 3\alpha-2x \right\}.$
I'm asked to find the volume of this set, which is $V_\alpha=\iiint_{E_\alpha}...
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0answers
44 views
Line element to polar coordinates
I'm calculating the effective metric for a vortex in polar coordinates. The velocity and the potential is:
\begin{equation}
\mathbf{v}=\frac{A}{r} \hat{r} + \frac{B}{r}\hat{\theta}
\end{equation}
So:...
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0answers
36 views
Cartesian coordinates to Cylindrical coordinates
The position vector, as a function of time and in Cartesian coordinates, of a particle is the following:
$$\vec{r}(t) = (5t^2-6t)\vec{e}_1 + \cos(\sin(t))\vec{e}_2+ (8t-5)\vec{e}_3$$
I have that ...
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0answers
21 views
Find the volume enclosed between cone and rose petal
I understand that while changing an integral from cartesian to cylindrical or polar or spherical coordinates requires multiplication of a Jacobian to integration variables but what when the region is ...
2
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0answers
120 views
Unwrapping a cylinder onto a plane
I have an image of a label on a bottle, and I need to 'unwrap' the image as if I had taken the label off of the bottle and laid it flat on a table.
Is there an easy way to transform the image, given ...
3
votes
4answers
129 views
How to evaluate the integrals in the cylindrical coordinates
Evaluate the following integral in cylindrical coordinates
$$\int^{1}_{-1}\int^{\sqrt{1-x^2}}_{0}\int^{2}_{0}\dfrac{1}{1+x^2+y^2}dzdydx$$
My try:
I first took the boundaries as $$-1\le x\le1\\0\le ...
2
votes
2answers
48 views
Conversion of a Vector in a Cartesian Coordinate System to a Cylindrical Coordinate System
I'm having trouble converting a vector from the Cartesian coordinate system to the cylindrical coordinate system (second year vector calculus)
Represent the vector $\mathbf A(x,y,z) = z\ \hat i - ...
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0answers
78 views
How to write basis vectors in Cartesian coordinates in terms of cylindrical coordinates?
Given a Cartesian coordinate system with basis vectors (ex, ey, ez)
and a Cylindrical coordinate system with basis vectors (er, eθ, ez)
Why and how does:
ex = erCos(θ) - eθSin(θ) ?
ey = erSin(θ) +...
1
vote
1answer
19 views
Describing this solid in cylindrical coordinates
Let $Q$ be the solid delimitated by the paraboloid $z = x^2 + y^2$, by the cylinder $x^2 + y^2 = 4x$ and by the plane $z = 0$. In cartesian coordinates, we can write: $$Q = \{ (x, y, z) \in \mathbb{R}^...
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votes
1answer
14 views
Converting the coordinates of a point to cylindrical coordinates with positive values.
I'm trying to convert the coordinates of point $(x,y,z) = (-2,-1,0)$ to cylindrical coordinates, with positive values for $\theta$ and $r$.
I know that:
$r^2 = x^2 + y^2$
so...
$r = \sqrt(5)$
...
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2answers
217 views
inward or outward normal to a surface?
I've got a conceptual problem regarding inward and outward normals. The textbook question (2nd year vector calculus) is as follows:
A uniform fluid that flows vertically downward is described by ...
-1
votes
3answers
60 views
Integrating $ \frac{1}{\sqrt {x^{2}+y^{2}}}-\frac{1}{\sqrt {y^{2}+z^{2}}}\ $
I am trying to solve this integral over a Cube:
$$\frac{1}{\sqrt {x^{2}+y^{2}}}-\frac{1}{\sqrt {y^{2}+z^{2}}}\
$$
I can see that this that this will turn into zero because of symmetry since I am ...
1
vote
1answer
57 views
How do I revolve a general 2D coordinate system?
$\newcommand{\dd}{\partial}$
Question
I wish to construct a general 3D revolved, orthogonal, curvilinear co-ordinate system, where the axis of revolution is coincident with the Cartesian $z$-axis. ...
0
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1answer
22 views
Volume using cylindrical coordinates
I have to find the volume of the solid which base is bounded by
$$x^{2}+y^{2}+2y=0$$
and it's bounded, above, by the surface
$$z=4-x^{2}-y^{2}$$
I tried to use cylindrical coordinates, where
$$x=r\...
0
votes
1answer
23 views
Compute an indefinite integral through change of variables
I'm trying to compute the following integral:
$\int_{-\infty}^\infty \int_{-\infty}^\infty e^{x_1^2 + x_2^2} dx_1 dx_2$
I have been thinking about a change of variables with cylindric coordinates, ...
2
votes
1answer
52 views
How to draw lines and circles on cylindrical projection map?
I am trying to draw a circle with known radius around a coordinate on a cylindrical projection map. Which is a circle around equator and egg shaped closer to the poles.
And also trying to draw a line ...
0
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0answers
43 views
Interpreting Cylindrical equations gemotrically
I'm a bit unsure how to interpret the following equation given in cylindrical coordinates geometrically.
tanθ = 1
From what I understand, this means that the angle b/w the x-axis and point in the ...
1
vote
1answer
345 views
Divergence of $1/r$ in cylindrical coordinates
In classical textbooks, like "Introduction to Electrodynamics" by J.D. Griffiths, it is given that
$$\nabla\cdot\left(\frac{\widehat{r}}{r^2}\right)=4\pi\delta^3(R).$$
To prove this equality, ...
0
votes
1answer
36 views
Perpendicular vector fields in cylindrical coordinates
With two vector fields in cylindrical coordinates, I am trying to find how they may be perpendicular to each other
$$ A (\rho, \phi,z) = \rho cos \phi \hat\rho + \rho sin \phi \hat\phi + \rho \hat z $...
0
votes
0answers
37 views
Derivative of vector in cylindrical coordinates with regards to $\varphi$?
We have cylindrical coordinates $(\rho,\varphi,z)$ with basis vectors $\hat e_\rho, \hat e_\varphi, \hat e_z$.
Let's say we have a vector $\vec v=z\hat e_\rho + \hat e_\varphi\ + \rho \hat e_z$.
How ...
3
votes
2answers
57 views
(Multivariable Calculus) Convert $\rho = \sin \phi$ to cylindrical and rectangular
Question: Consider the surface given in spherical coordinates by $\rho = \sin(\phi)$. Convert to rectangular coordinates and cylindrical coordinates. Identify the surface.
By graphing the function, I'...
0
votes
2answers
112 views
How do we deal with ArcTan (or other inverse functions) of undefined values?
When converting coordinates from rectangular to cylindrical, to spherical, etc. we will eventually come across having to use $ArcTan(y/x)$, and/or $ArcCos(z/\rho)$ to derive $\theta$ and/or $\phi$. ...
0
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1answer
83 views
Representing displacement vectors in cylindrical coordinates and finding the distance in cylindrical coordinates?
In cartesian coordinates, we can derive the vector $\vec v_3$ by vector subtraction $\vec v_2-\vec v_1$. We then get the distance between $P$ och $Q$ by taking the absolute value of $\vec v_3$ which ...
