Small question about derivatives and gradients.

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SUMMARY

The discussion clarifies that the gradient of the partial derivative of a function \( f \) with respect to \( t \), denoted as \( grad\left(\frac{\partial f}{\partial t}\right) \), is equivalent to the partial derivative of the gradient of \( f \) with respect to \( t \), expressed as \( \frac{\partial}{\partial t}(grad f) \), under standard conditions. This relationship is similar to the equality of mixed partial derivatives, \( \frac{\partial^2 f}{\partial x \partial t} = \frac{\partial^2 f}{\partial t \partial x} \), which holds true unless \( f \) exhibits unusual characteristics.

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Is the grad([itex]\frac{\partial f}{\partial t}[/itex]) the same as [itex]\frac{\partial}{\partial t}[/itex](gradf)?

Thank you.
 
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Hi M. next! :wink:
M. next said:
Is the grad([itex]\frac{\partial f}{\partial t}[/itex]) the same as [itex]\frac{\partial}{\partial t}[/itex](gradf)?

Unless f is really weird, yes. :smile:

(it's the same test as whether ∂2f/∂x∂t = ∂2f/∂t∂x …

i forget the exact conditions)​
 
Thank you!
 

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