When is a vector field is the gradient field of a scalarfunction?
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From: Bastian Erdnuess on 18 May 2010 15:29 Andrew Magyar wrote: > From vector calculus, in 3-dimensions it is a known fact that a given > vector field is the gradient field of scalar function from R^3 -> R if > the curl of that vector field is equal to 0. > > Correct me if I am wrong, but this condition really boils down to the > fact that the order in which second order partial derivatives is taken > is irrelevant (ie d^2f/dxdy = d^2f/dydx). No. That fact explains why the curl of a gradient field is zero. But it doesn't tell why it is sufficient to check that the curl of a vector field is zero to be sure that it is a gradient field. > In general d-dimensions, given a vector field the question of > interests for me is whether this given vector field corresponds to the > gradient field of a scalar function from R^d -> R. > > Is the condition for which this is true similar to that in R^3, namely > that the order in which you take second order partial derivative be > irrelevant? Indeed, it is sufficient to check from a vector field f whether d/dx_i f_j = d/dx_j f_i or not to know whether it is a gradient field or not. That condition is obviously true for gradient fields, but it is not so obvious that this condition is a sufficient condition for a vector field to be a gradient field. Cheers, Bastian |