The jacobian matrix has some really interesting properties when applied to surface and volume geometry. What is the formula that would give out the surface/volume element of any dimensional object? As example let's consider calculating the jacobian of a 26 dimensional sphere. Calculating this by hand would take years and without the help of a general formula for spherical determinant, it would be quite hard. Another problem comes out, what is the proof for this kind of formula? Lets start by writing an n by n square matrix.
\left[ {\begin{array}{*{20}{c}} {\frac{{\partial {x_1}}}{{\partial {\theta _1}}}}&{\frac{{\partial {x_1}}}{{\partial {\theta _2}}}}&{\frac{{\partial {x_1}}}{{\partial {\theta _3}}}}&{\frac{{\partial {x_1}}}{{\partial {\theta _4}}}}& \ldots &{\frac{{\partial {x_1}}}{{\partial {\theta _n}}}}\\ {\frac{{\partial {x_2}}}{{\partial {\theta _1}}}}&{\frac{{\partial {x_2}}}{{\partial {\theta _2}}}}&{\frac{{\partial {x_2}}}{{\partial {\theta _3}}}}&{\frac{{\partial {x_2}}}{{\partial {\theta _4}}}}& \ldots &{\frac{{\partial {x_2}}}{{\partial {\theta _n}}}}\\ {\frac{{\partial {x_3}}}{{\partial {\theta _1}}}}&{\frac{{\partial {x_3}}}{{\partial {\theta _2}}}}&{\frac{{\partial {x_3}}}{{\partial {\theta _3}}}}&{\frac{{\partial {x_3}}}{{\partial {\theta _4}}}}& \ldots &{\frac{{\partial {x_3}}}{{\partial {\theta _n}}}}\\ {\frac{{\partial {x_4}}}{{\partial {\theta _1}}}}&{\frac{{\partial {x_4}}}{{\partial {\theta _2}}}}&{\frac{{\partial {x_4}}}{{\partial {\theta _3}}}}&{\frac{{\partial {x_4}}}{{\partial {\theta _4}}}}& \ldots &{\frac{{\partial {x_4}}}{{\partial {\theta _n}}}}\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ {\frac{{\partial {x_n}}}{{\partial {\theta _1}}}}&{\frac{{\partial {x_n}}}{{\partial {\theta _2}}}}&{\frac{{\partial {x_n}}}{{\partial {\theta _3}}}}&{\frac{{\partial {x_n}}}{{\partial {\theta _4}}}}& \cdots &{\frac{{\partial {x_n}}}{{\partial {\theta _n}}}} \end{array}} \right]
The determinant of a matrix can be obtain from the sum of all the product of the right-oriented diagonals then subtract the sum of all the product of the left-oriented diagonals. The first line of diagonal is made completely of matching numbers, X 1, Theta 1, X 2 Theta 2, so it's easy to find the rule for this one. When it comes to the other diagonals, numbers are following another order each time, the formula I found to calculate it is this:
This is where my interrogation comes from. How do I prove this formula? It seems very complex and I have no reference or any idea how to prove it. I tested in a 3 dimensional space with coordinates change from Cartesian to spherical to compute the volume of a sphere and it worked without any problem. I'm wondering if anyone have an idea how to prove it or if it's wrong, I would like to know where the equation breaks.
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That method of taking the sum of all the products of the right-oriented diagonals minus the sum of all the products of the left-oriented diagonals only works for 1×1, 2×2, and 3×3 matrices.
You can verify that ∣∣∣∣∣∣∣∣2100120000210012∣∣∣∣∣∣∣∣=9, but that method yields 15 as the determinant.
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I didn't know that it only worked for 2 and 3 dimension matrix. The other method is like splitting 1 matrix up with smaller ones with the sign matrix. Thanks for making me realize this. I will try to compute a general formula using the sum of the product of the determinant of smaller matrix.
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The determinat of an n×n matrix, whose (i,j)th component is ai,j, is π∈Sn∑sgn(π)a1,π(1)×a2,π(2)×a3,π(3)×⋯×an,π(n) where the sum is over all permutations π of n symbols, and sgn(π) is the sign of the permutation. Every permutation can be written as a product of transpositions (permutations that just swap two elements), and the parity (evenness/oddness) of the number of transpositions required is fixed for each permutation. We say that sgn(π)=1 if an even number of transpositions is required, and −1 if an odd number is needed.
Each term a1,π(1)×a2,π(2)×⋯×an,π(n) is one of the n! ways of simultaneously taking one element from each row and each column, and multiplying them together. The sgn(π) term tells whether you should add or subtract it in.
I am afraid you are not going to find any much simpler general formula than this.