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Multi-Variable Function


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A function of 10 variables in addition to that of time, have a gradient of F(x1, x11,t)! Determine the gradient properties. Away from the constrictions on this. I think that integrating F on any path is independent upon the velocity-properties. Can any one advise!

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A function of 10 variables in addition to that of time, have a gradient of F(x1, x11,t)! Determine the gradient properties. Away from the constrictions on this. I think that integrating F on any path is independent upon the velocity-properties. Can any one advise!

 

You have a very much undefined and unconstrained problem statement.

 

The line integral of the gradient of any differentiable function over a closed loop is 0. Other than that your problem is wide open.

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I didn't say closed loop!

 

The integral around every closed loop being zero is equivalent to the integral over an arbitrary path depending only on the end points of the path.

 

You started by stating that you vectorvfield arose as a gradient. Potential fields are closed, therefore exact. Your field is necessarily conservative; i.e. line integrals are independent of path.

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But, it is dynamic and also conservativity is only for 3 dimensions not for 11 [Do you have a curl definition for more than 3 dimensions?].

 

I don't need a curl definition for more than three dimensions. The integral of a gradient around a closed curve is still zero.

 

You need to be more clear about what you mean by "dynamic" and how your function is defined. All that you have said thus far is that you have a gradient and have chosen to call one variable "t".

 

The 3-dimensional vector analysis to which you refer extends to analysis in any dimension and to manifolds of arbitrary dimension. The subject is differential geometry. You might want to take a look at Mike Spivak's little book Calculus on Manifolds.

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