 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/1/a/solution

The three coordinate basis vectors are the partial derivatives of the position vector with respect to the three coordinates. Direct differentiation gives

h_{ξ} = \frac{\partial x}{\partial ξ} = cosh ξ sin θ cos ϕ cosh ξ sin θ sin ϕ g^{'} (ξ) cos θ, h_{θ} = \frac{\partial x}{\partial θ} = sinh ξ cos θ cos ϕ sinh ξ cos θ sin ϕ - g (ξ) sin θ,

h_{ϕ} = \frac{\partial x}{\partial ϕ} = - sinh ξ sin θ sin ϕ sinh ξ sin θ cos ϕ 0 .

The multivariable chain rule then gives

d x = h_{ξ} d ξ + h_{θ} d θ + h_{ϕ} d ϕ

. These are coordinate tangent vectors, not necessarily unit vectors.

 New to topics? Read the docs here!