Exercise 2.06 Helix and tangent vector in Cartesian and spherical polar

Question

Consider R³ as a manifold with the flat Euclidean metric, and coordinates {x, y, z}. Introduce spherical polar coordinates {r, θ, ϕ} related to {x, y, z} by

x = rsin⁡θcosϕ⁡ (1)

y = rsin⁡θsin⁡ϕ (2)

z = rcos⁡θ (3)

so that the metric takes the form

ds² = dr² + r²dθ² + r²sin²θdϕ² (4)

(a) A particle moves along a parameterised curve given by

x(λ) = cos⁡λ  ,  y(λ) = sin⁡λ  ,   z(λ) = λ (5)

Express the path of the curve in the {r, θ, ϕ} system.

(b) Calculate the components of the tangent vector to the curve in both the Cartesian and spherical polar coordinates.

Answers

It is fairly obvious that the curve is a helix. It has unit radius, the distance between each rung is 2π and so it goes up at an angle of 45°. It is shown below, compressed in the Z direction.

The helix, with some vector components

Calculating the components of the tangent vector d/dλ in polar coordinates was non-trivial for me. It involved the chain, the quotient and the cos^-1 rules of differentiation. I checked them by using the tensor transformation law from Cartesian to spherical polar components. The law is

In this case b…z and α...ω indices disappear, so it became a bit simpler but there were still almost 50 equations to step through and I had to add the tan^-1 rule of differentiation to my armoury. I discovered that this method gave a different result for dθ/dλ component of my tangent vector. The possibility of errors had became enormous. Symbolab, a wonderful differential equation calculator, came to my rescue and I used it to check everything. It discovered a couple of minor errors in my 50 equation epic, but eventually pinned down the error to the first calculation of dθ/dλ from the equation of the curve in the {r, θ, ϕ} system. It just goes to show how important it is to have an 'independent' check.

More Gains

1) I have also now realised why the metric is sometimes written in the form like

ds² = dr² + r²dθ² + r²sin²θdϕ²

and sometimes as a matrix. And how to get from one to the other.

2) In oder to draw the helix, I developed a spreadsheet to draw the 3-D curve. I am inspired to do something more general to draw any 3-D curves from a specified perspective.

The full 11 page answer is at Ex 2.06 Helix.pdf. It includes a reference to the spreadsheet and from page 6 is mostly Differentiation in baby steps and other detailed calculations.