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Questions tagged [curvilinear-coordinates]

Use this tag for questions about coordinate systems for Euclidean space for which coordinate lines may be curved.

3
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1answer
65 views

Why are these two ways of evaluating the curl of a vector field not actually equivalent?

I was asked to calculate the curl of the vector field $\theta^2\bf e_\theta$ in spherical coordinates. If we use the typical formula for curl in curvilinear coordinates: $$ \nabla\times \bf F = \left|\...
2
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1answer
42 views

Limit of Laplacian of the distance at the origin

Let $p$ be a point in a Riemannian manifold $M$ and $d_p$ be the distance from the point $p$. Prove that $\lim_{x\rightarrow p}\Delta d_p(x)=\infty$ I can easily prove it in $\mathbb{R}^n$. But for a ...
2
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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 - ...
2
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1answer
35 views

Trying to prove (a x (b x c)) = b (a.c) - c (a.b)

I'm trying to prove that $$ \mathbf{ a }\times (\mathbf{b} \times \mathbf{c}) = \mathbf{b} ( \mathbf{a} \cdot \mathbf{c} ) - \mathbf{c} ( \mathbf{a} \cdot \mathbf{b} ), $$ using the identity $$ \...
2
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0answers
52 views

Identity in vector calculus : why is it zero?

From Spiegel's "Vector Analysis", problem 7.21 a): We have general curvilinear coordinates, $u_i$, with unit vectors $\mathbf{e}_i$ for $i=1,2,3$ and a vector $\mathbf{A}=A_1\mathbf{e}_1+A_2\mathbf{e}...
1
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1answer
45 views

Coordinate and Non-coordinate basis (Orthonormality and Orthogonality)

I am having a really hard time in understanding the difference between coordinate and non-coordinate basis. I tried looking for some relevant questions here but none answers my concerns, may be my ...
1
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1answer
28 views

Define the path integral when scalar $f$ and curve $\mathbf{c}$ is in *curvilinear* coordinates

I'm doing a multivariable calculus course at the moment. I've seen path integrals in the cartesian coordinate system as the following definition: Definition. The path integral of $f(x,y,z)$ along the ...
1
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1answer
51 views

Coordinate basis and coordinate systems

When we introduce coordinate systems, like spherical coordinates, one usually does it with respect to cartesian coordinates. What would be the right way to derive the (for example) spherical ...
1
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1answer
29 views

Calculating the Weight of a Coil of Sheet Metal

At work we use generic tables to estimate the weight of coils of sheet metal. The tables are based on the thickness in inches of metal "sidewall" that remains on the coil. Now upper management says ...
1
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0answers
46 views

How do I generalize the dot product (bilinear form) in spherical coordinates?

In cartesian coordinates, the unit vectors $\{u_x, u_y, u_z\}$ are universal. That is, $u_x(x, y, z)$ is constant and so on for the rest of them. Because of that, the dot product $\langle v | w \...
1
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1answer
24 views

How to find inverse of general curvilinear coordinates

Lets say I have a curvilinear coordinate system $A=A(x,y,z) = \frac{x^2+y^2+z^2}{2z} $, $B=B(x,y,z)= \frac{x^2+y^2+z^2}{2\sqrt{x^2+y^2}}$, $C=C(x,y,z)=\tan^{-1}(y/x)$ How do I find the inverse of ...
1
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1answer
34 views

How do I derive the volume element $ dV = \sqrt{g} du^1 du^2 du^3 $ in a 3D curvilinear coordinate system?

I am trying to derive $ dV = \sqrt{g} du^1 du^2 du^3 $ for some general curvilinear coordinate $(u^1,u^2,u^3)$ system in $\mathbb{R}^3$ where $g = \mathrm{det}[g_{ij}]$. I am using the following facts:...
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0answers
96 views

Scale factors in orthogonal coordinate systems

I'm trying to reduce the general tensor expression for either orthogonal or non orthogonal $$\vec{\nabla} \cdot \vec{V} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial x_i}\left(\sqrt{g}V^i \right)$$ ...
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0answers
38 views

Find The Area of the Surface $z=xy$ for $x^2+y^2\leq 4$

This question comes under the topic of orthogonal curvilinear coordinates, but I am unsure how that topic relates to this question. How should I approach this question? This content will be covered ...
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0answers
8 views

Finding curl$\vec F$ of the field $\vec F=f(\phi,\theta)e_\rho$

Find curl$\vec F$ in spherical coordinates for a vector field of the form $\vec F=f(\phi,\theta)e_\rho$. My query is that I have not seen a vector field in this form before. Can someone explain it to ...
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0answers
45 views

Transform vectors from cartesian coordinates to curve coordinate system [2d]

I have an object moving in a two-dimensional space and its position is given by cartesian coordinates $(x_i, y_i)$. This object also has a velocity vector $({v_x}_i,{v_y}_i)$ and an acceleration ...
1
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0answers
35 views

Given $\alpha(x,y,z) = \alpha_1\hat x +\alpha_2 \hat y+\alpha_3 \hat z$, how to write in the coordinates $\hat r, \hat \theta, \hat \phi$

I'm trying to convert a parametrization of a curve $$\vec \alpha(x,y,z) = \alpha_1(x,y,z) \hat x + \alpha_2(x,y,z) \hat y + \alpha_3(x,y,z) \hat z$$ to $$\vec \alpha(r, \theta, \phi) = \alpha_r(r, \...
1
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1answer
129 views

On the relationship between the volume element in curvilinear coordinates and the Jacobian

Suppose we have a curvilinear coordinate system in $\mathbb{R}^3$, $(q_1, q_2, q_3)$, with $x_i = f_i(q_1, q_2, q_3)$. We define the local basis for this coordinate system by $\hat{e_i} = \frac{\...
0
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1answer
30 views

Why is the normal vector different in cartesian coordinates vs. spherical coordinates?

Consider the sphere $x^2+y^2+z^2=1$. Let $\mathbf x(u,v)$ be a parameterization for the sphere. Say I was trying to find specifically the normal vector given by $$ \frac{\partial \bf x}{\partial u} \...
0
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1answer
170 views

Find the center of mass in 3D

Vector Calculus: [Using Integration] Find the center of mass of the "snow cone" of uniform density bounded above by the sphere $x^2+y^2+z^2=a^2$ and below by the cone $z=\sqrt{x^2+y^2}$.
0
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1answer
873 views

Dot product in spherical coordinates

What can be the formula for dot product in spherical coordinates? that is $$\overrightarrow{A}(r, \theta , \phi). \overrightarrow{B}(r, \theta , \phi)=?$$ $$or,(A_r \hat{r} + A_{\theta} \hat{\theta} + ...
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0answers
16 views

Question on the derivation of Laplace operator, and its application to polar coordinate system.

Let $\mathbf r = \left[ \begin{matrix} r & \phi \end{matrix} \right]^\top \; $be some curvilinear coordinates, with corresponding unit base column vectors $\hat {\mathbf h}_r \; $and $\hat {...
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0answers
24 views

Curvilinear Coordinates Transform matrix

In Kusse's Mathematical Physics, equation gives the transformation matrix between 2 curvilinear systems as $q'_i=a_{ij}q_j$. Equation 4.100 lists $a$ as: $$a_{ij}=\frac{h'_i}{h_j}\frac{\partial q_i'}{\...
0
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1answer
16 views

Name of Non-Unique Coordinate systems

Are there some examples (and a name) for non-unique coordinates (non-unique meaning may have multiple ways to represent the same point). Such as the one below.
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0answers
31 views

Is this a correct definition of a Riemannian metric tensor?

I am a total beginner in Riemannian geometry but I'm trying to teach myself the basics. So the following could contain many horrible mistakes. Suppose I describe the x,y-plane by curvilinear ...
0
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0answers
21 views

Curvilinear abscissa = radius * angle - Circular motion

I would like to understand why: $$ s(t) = r \, \theta(t) $$ where $s$ is the curvilinear abscissa, $r$ the radius and $\theta$ the angle in circular motion. Thank you for your time.
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0answers
40 views

Del operator apply directly to orthogonal curvilinear coordinate does not match

I understand that $\nabla$ in general orthogonal coordinate $(u_1,u_2,u_3)$ as follows: $$ \nabla=\mathbf{a}_{u_1}\frac{\partial}{h_1\partial u_1}+\mathbf{a}_{u_2}\frac{\partial}{h_2\partial u_2}+\...
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0answers
104 views

Divergence in Orthogonal Curvilinear Coordinates - Is there a mistake?

I have to prove that, $$\nabla \cdot (A_1\mathbf e_1) = \frac{1}{h_1h_2h_3}\frac{\partial (A_1h_2h_3)}{\partial u_1} $$ My approach: $$\begin{align} \nabla \cdot (A_1\mathbf e_1) &= \nabla \...
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0answers
14 views

Benjamin's Othogonal Curvilinear Coordinate System to analyze Gas Velocity

I have been trying to understand a problem given in a paper for a couple of months but cannot figure out the rationale behind the change of variables of a function. This problem is outlined below. In ...
0
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1answer
58 views

How to derive the transformation that maps general curvilinear coordinates to their cartesian image.

Assume an arbitrary orthogonal curvilinear coordinate system: $$ \mathbf θ \in \mathrm A \subset \mathbb R ^2 \leftrightarrow \mathbf x \in \mathbb R ^2 $$ Notice the following notation that I use: $$...
0
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2answers
110 views

Path on a sphere

I am trying to solve a exercise problem in GR on a "triangle" whose sides are great circles of a sphere of radius $R$. So this is the triangle that I chose (coordiantes are written as $(r, \theta, \...
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0answers
213 views

Inverse of a (coordinate) transformation - which method is best?

I am woring with curvilinear coordinates and got stuck on an exercise where we have a transformation from cartesian to parabolic coordinates $(x,y,z)\rightarrow (u,v,\varphi)$. My question is how I ...
0
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0answers
53 views

Invariants of a general 2nd order tensor

I am attempting to write out three invariants of a general (not necessarily symmetric or Cartesian), 2nd order tensor which we will call $\mathbf{T} = T^i_{\;j}\mathbf{e}_i\mathbf{e}^j \,, \;\; i,j=1,...
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0answers
32 views

Formula for a Line Integral in Curvilinear Coordinates.

I am familiar with the formula for a path integral given a parametrisation $\textbf{x}(t)$, $t \in l \subseteq \mathbb{R}$ of a curve $C$, and given some scalar function $f(x,y,z)$, $$\int_C f ds = \...
0
votes
1answer
179 views

Expressing contravariant basis vectors in terms of position vector

I am learning about general curvilinear bases for my research, and I have a question about notation for contravariant basis vectors. Suppose we have a point in space defined by the position vector $\...
-1
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1answer
34 views

Proof of formula for curvilinear curl

I'm trying to prove the formula for curl in curvilinear coordinates and am following the proof in the book "Vector Analysis and Cartesuan Tensors" by Bourne and Kendall. I got this far: -rewritting $...