## Question

Consider**R**³ 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*=

*r*sin

*θ*cos

*ϕ* (1)

*y = r*sin

*θ*sin

*ϕ*(2)

z =

so that the metric takes the form*r*cos*θ*(3)
d

*s*^{2}= d*r*^{2}+*r*^{2}d*θ*^{2}+*r*^{2}sin^{2}*θ*d*ϕ*^{2}(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
d

*s*^{2}= d*r*^{2}+*r*^{2}d*θ*^{2}+*r*^{2}sin^{2}*θ*d*ϕ*^{2}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.

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