youngurlee
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I have shown by my intuition that if a good field g(2th or more differentiable) in n dimension satisfies
\frac{∂g_{i}}{∂x_{j}}-\frac{∂g_{j}}{∂x_{i}}=0 for all i,j,
then \ointg\cdotdl=0,
hence there exist a scalar function \phi such that
\frac{∂\phi}{∂x_{i}}=g_{i} for all i.
I want to know what in general will the ring integral \ointg\cdotdl be,
can it be written as a surface integral, as in Kelvin–Stokes theorem?
\frac{∂g_{i}}{∂x_{j}}-\frac{∂g_{j}}{∂x_{i}}=0 for all i,j,
then \ointg\cdotdl=0,
hence there exist a scalar function \phi such that
\frac{∂\phi}{∂x_{i}}=g_{i} for all i.
I want to know what in general will the ring integral \ointg\cdotdl be,
can it be written as a surface integral, as in Kelvin–Stokes theorem?