4
votes
0answers
145 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
56 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.
0
votes
0answers
56 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
95 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
412 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
81 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 $M$ ...
6
votes
0answers
402 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
197 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
148 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
244 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
279 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
96 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 ...