Derivatives of coordinate conversions

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Theory

Cartesian to spherical

The Cartesian to spherical conversion is defined by:

`f_{text(hyperspherical)}^-1 : RR^n -> RR_+ xx text([)0, pi text([)^(n - 2) xx text([) 0, 2 pi text([) : f_{text(hyperspherical)}^-1(x_1, ..., x_n) = ( (sqrt(x_1^2 + ... + x_n^2)), (\vdots), (atan2(sqrt(x_i^2 + ... + x_n^2), x_(i - 1))), (\vdots), (atan2(x_n, x_(n - 1))))`

The partial derivative of this function w.r.t. `x_k` can be evaluated based on the following results:

`d / dx_k sqrt(x_1^2 + ... + x_n^2) = x_k / sqrt(x_1^2 + ... + x_n^2)`

`d / dx_k text(atan2)(sqrt(x_i^2 + ... + x_n^2), x_{i - 1}) = [ [(x_k x_{i - 1}) / (sqrt(x_i^2 + ... + x_n^2) * (x_{i - 1}^2 + ... + x_n^2)), text(, ) k in \[i, n\]], [-sqrt(x_i^2 + ... + x_n^2) / (x_{i - 1}^2 + ... + x_n^2), text(, ) k = i - 1], [0, text(, otherwise)]]`

`d / dx_k text(atan2)(x_n, x_{n - 1}) = [ [-x_n / (x_{n - 1}^2 + x_n^2), text(, k = n - 1)], [x_{n - 1} / (x_{n - 1}^2 + x_n^2), text(, k = n)], [0, text(, otherwise)]]`

Spherical to Cartesian

The spherical to Cartesian conversion is defined by:

`f_{text(hyperspherical)} : RR_+ xx text([)0, pi text([)^(n - 2) xx text([) 0, 2 pi text([) -> RR^n : f_{text(hyperspherical)}(r, alpha_1, ..., alpha_{n - 1}) = ( (r text(cos)(alpha_1)), (r text(sin)(alpha_1) text(cos)(alpha_2)), (\vdots), (r text(sin)(alpha_1) \cdots text(sin)(alpha_{n - 2}) text(cos)(alpha_{n - 1})), (r text(sin)(alpha_1) \cdots text(sin)(alpha_{n - 2}) text(sin)(alpha_{n - 1})))`

Remark 1.7.5 - Page generated 16.06.2017 01:36.