3
votes
0answers
30 views

What distinguishes elliptical coordinates from polar coordinates?

I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions. Polar: $$ x=r \cos(t) \\ y=r \sin(t) $$ ...
1
vote
2answers
50 views

Global and local coordinates on a manifold, and their relations to curvature

I would be pleased to have some information about coordinates in differential geometry. A) First I would like to check whether or not the definitions I use are correct. (Mainly for the sake of ...
1
vote
1answer
60 views

Definition of smooth maps between manifolds

Here is a page from Guillemin-Pollack's differential topology: My question is: At the bottom he defines $df=d\psi\circ dh\circ d\phi^{-1}$. Why doesn't he just define $df=dh$, like here: ...
5
votes
0answers
165 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
2
votes
3answers
58 views

What separates rotations from other co-ordinate transformations?

I am confused about some seemingly elementary ideas. From what I have understood, a rotation is just a specific class of co-ordinate transformations. If this is true, what exactly separates a rotation ...
0
votes
1answer
45 views

Geometry finding area problem

A regular 2N -sided polygon of perimeter L has its vertices lying on a circle. Find the radius of the circle and the area of the polygon.
4
votes
0answers
182 views

Orthogonal Coordinate Systems Intuition

I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, ...
3
votes
0answers
46 views

Can all 2-surfaces be “coordinated” using 2 numbers

Consider a 2-dimensional surface embedded in 3 dimensional euclidean space. ex: A plane, a sphere, a hyperboloid of 2 sheets (or 1 sheet), the graph of sin(x + y), 2 parallel planes etc... If we ...
1
vote
1answer
96 views

Coordinate transform

Can anyone see what transformation $$r\to f(r)$$ transforms $$\exp(2\phi(r))(dr^2+r^2d\theta^2)$$ to $$df^2+\sinh^2(f)d\theta^2$$? I there a systematic way to attack such a problem -- rather than just ...
3
votes
1answer
499 views

Computing gradient in cylindrical polar coordinates using metric?

I am trying to understand coordinate transformations properly (having studied some general relativity in the past). Let us consider the transformation from cartesian to cylindrical coordinates, ...
1
vote
1answer
90 views

Zero Locus of Functions is a Submanifold

Suppose $f_1,\dots, f_d$ are a set of real-valued functions on a smooth manifold $M$. Let $N$ be the zero locus of the $f_i$. Suppose the $\textrm{d}f_i$ span a subspace of the cotangent space of $M$ ...
7
votes
0answers
420 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
1
vote
1answer
207 views

Gradient in curvilinear coordinates

The gradient is usually written as the product of the unit vectors times the derivative with respect to that coordinate. In Einstein summation convention: $\hat e_i \partial_i$ I've seen it written ...
6
votes
4answers
3k views

Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that i have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
1
vote
1answer
154 views

Curves and first fundamental form

Would I be right to think that if I have a coordinate system $(x,y)$ so that the lines/curves where one coordinate is fixed, so something like $x=a$ and $y=b$, always intersect at the same angle, then ...
0
votes
1answer
257 views

Rigid motion in curvilinear coordinates

I would like someone to clarify this since it has bedazzled me and can't seem to get a grip on it. Consider a 3D real space and Euclidean coordinates ($x_1,x_2,x_3$), with an associated standard basis ...
4
votes
2answers
314 views

How do I convert a vector field in Cartesian coordinates to spherical coordinates?

I have a vector field in terms of $\mathbf{\hat i}$, $\mathbf{\hat j}$, and $\mathbf{\hat k}$, $$\mathbf{F} = x\mathbf{\hat i} + y\mathbf{\hat j} + z\mathbf{\hat k}$$ How do I convert it to the ...
3
votes
0answers
100 views

Translating coordinates on a Riemann surface

Let $U\subset X$ be an open subset of a connected Riemann surface $X$. Let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. Let $P\in U$ be the ...