Cramer's rule

Cramer's rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. It is named after Gabriel Cramer (1704 - 1752).

Applications to differential geometry

Cramer's rule is also extremely useful for solving problems in differential geometry. Consider the two equations F(x, y, u, v) = 0, and G(x, y, u, v) = 0,. When u and v are independent variables, we can define x = X(u, v), and y = Y(u, v),.

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Finding an equation for partial x/partial u is a trivial application of Cramer's rule.

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First, calculate the first derivatives of F, G, x and y.

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:dF = rac{partial F}{partial x} dx + rac{partial F}{partial y} dy +rac{partial F}{partial u} du +rac{partial F}{partial v} dv = 0

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:dG = rac{partial G}{partial x} dx + rac{partial G}{partial y} dy +rac{partial G}{partial u} du +rac{partial G}{partial v} dv = 0

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:dx = rac{partial X}{partial u} du + rac{partial X}{partial v} dv

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:dy = rac{partial Y}{partial u} du + rac{partial Y}{partial v} dv

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Substituting dx, dy into dF and dG, we have:

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:dF = left(rac{partial F}{partial x} rac{partial x}{partial u} +rac{partial F}{partial y} rac{partial y}{partial u} +rac{partial F}{partial u} ight) du + left(rac{partial F}{partial x} rac{partial x}{partial v} +rac{partial F}{partial y} rac{partial y}{partial v} +rac{partial F}{partial v} ight) dv = 0

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:dG = left(rac{partial G}{partial x} rac{partial x}{partial u} +rac{partial G}{partial y} rac{partial y}{partial u} +rac{partial G}{partial u} ight) du + left(rac{partial G}{partial x} rac{partial x}{partial v} +rac{partial G}{partial y} rac{partial y}{partial v} +rac{partial G}{partial v} ight) dv = 0

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Since u, v are both independent, the coefficients of du, dv must be zero. So we can write out equations for the coefficients:

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:rac{partial F}{partial x} rac{partial x}{partial u} +rac{partial F}{partial y} rac{partial y}{partial u} = -rac{partial F}{partial u}

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:rac{partial G}{partial x} rac{partial x}{partial u} +rac{partial G}{partial y} rac{partial y}{partial u} = -rac{partial G}{partial u}

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:rac{partial F}{partial x} rac{partial x}{partial v} +rac{partial F}{partial y} rac{partial y}{partial v} = -rac{partial F}{partial v}

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:rac{partial G}{partial x} rac{partial x}{partial v} +rac{partial G}{partial y} rac{partial y}{partial v} = -rac{partial G}{partial v}

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Now, by Cramer's rule, we see that:

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:

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rac{partial x}{partial u} = rac{egin{vmatrix} -rac{partial F}{partial u} & rac{partial F}{partial y} \ -rac{partial G}{partial u} & rac{partial G}{partial y}end{vmatrix}}{egin{vmatrix}rac{partial F}{partial x} & rac{partial F}{partial y} \ rac{partial G}{partial x} & rac{partial G}{partial y}end{vmatrix}}

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This is now a formula in terms of two Jacobians:

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:rac{partial x}{partial u} = - rac{left(rac{partialleft(F, G ight)}{partialleft(y, u ight)} ight)}{left(rac{partialleft(F, G ight)}{partialleft(x, y ight)} ight)}

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Similar formulae can be derived for rac{partial x}{partial v}, rac{partial y}{partial u}, rac{partial y}{partial v}.

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