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October 18th, 2019, 12:24 AM   #1
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Gradient and it's effect in math and statistics?

I have a very basic understanding of what a gradient is, but what is it's effect in math and statistics?

(I know this question is maybe to broad)
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October 18th, 2019, 02:21 AM   #2
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Math Focus: Elementary Math
Vectorial Analysis . https://en.wikipedia.org/wiki/Vector...tial_operators
For example $\displaystyle \nabla \vec{F} $ measures the rate and direction of change in a scalar field F.
also you may find the gradient in some books written as grad(f).
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Last edited by idontknow; October 18th, 2019 at 02:25 AM.
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October 18th, 2019, 04:46 AM   #3
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Physically, a number of processes depend on the gradient of a field (how quickly it changes in space, and in what direction).

For example, in heat conduction, the heat flux is proportional to the gradient of the temperature field $-\vec{\nabla}T$.
In diffusion of solutions, the diffusion flux is proportional to the gradient of the concentration field $-\vec{\nabla}C$.
Pressure gradients in fluids (or solids, for that matter) produce a force per unit volume $f=-\vec{\nabla}P$.

The gradient is also used in some numerical approximation methods for finding minima and maxima of functions, among other things.


Incidentally, you can also take the gradient of a vector field, although this gives you a second-order tensor. For example, the stress tensor in Newtonian fluids is proportional to $\vec{\nabla}\vec{u}$.
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