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If [itex]f:S \to R[/itex] is a continuous function on a surface S, we define

[tex]

\int\limits_{}^{} {\int\limits_S^{} {fdS} } = \int\limits_{}^{} {\int\limits_D^{} {\left( {fo\Phi } \right)} } \left\| {\frac{{\partial \Phi }}{{\partial u}} \times \frac{{\partial \Phi }}{{\partial v}}} \right\|dudv

[/tex]

where [itex]\Phi :D \subset R^2 \to R^3 [/itex] is a regular, 1-1 parameterisation of [itex]S = \Phi \left( D \right)[/itex].

The integral

[tex]\int\limits_{}^{} {\int\limits_S^{} {fdS} } [/tex] does not depend on the choice of parameterisation [itex]\Phi [/tex].

This follows from the change of variables formula for double integrals.

The formula for n dimension integrals is:

[tex]\int\limits_D^{} {f\left( {x_1 ,...,x_n } \right)} dx_1 ...dx_2 = \int\limits_{D^* }^{} {f\left( {h\left( {u_1 ,...,u_n } \right)} \right)\left| {\det Dh\left( {u_1 ,...,u_n } \right)} \right|} du_1 ...du_n [/tex]

where h(D*) = D.

[tex]

\int\limits_{}^{} {\int\limits_S^{} {fdS} } = \int\limits_{}^{} {\int\limits_D^{} {\left( {fo\Phi } \right)} } \left\| {\frac{{\partial \Phi }}{{\partial u}} \times \frac{{\partial \Phi }}{{\partial v}}} \right\|dudv

[/tex]

where [itex]\Phi :D \subset R^2 \to R^3 [/itex] is a regular, 1-1 parameterisation of [itex]S = \Phi \left( D \right)[/itex].

The integral

[tex]\int\limits_{}^{} {\int\limits_S^{} {fdS} } [/tex] does not depend on the choice of parameterisation [itex]\Phi [/tex].

This follows from the change of variables formula for double integrals.

The formula for n dimension integrals is:

[tex]\int\limits_D^{} {f\left( {x_1 ,...,x_n } \right)} dx_1 ...dx_2 = \int\limits_{D^* }^{} {f\left( {h\left( {u_1 ,...,u_n } \right)} \right)\left| {\det Dh\left( {u_1 ,...,u_n } \right)} \right|} du_1 ...du_n [/tex]

where h(D*) = D.

*It would make sense that the integral of f over S doesn't depend on parameterisation of the surface. But I don't see how that follows from the change of variables formula. (Similarly, an integral of a scalar function over a path doesn't depend on paramterisation of curve and this again follows from change of variables formula). Can someone explain why this is the case to me? Any help would be good thanks.*
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