Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a question that gives me a 3d function and asks me to calculate the gradient vector of it. This part I understand. It then asks me to indicate the points at which it does not exist. When does a gradient vector not exist? Is it when it equals to zero? Or does it mean when the function DNE.

Thanks.

share|improve this question
add comment

1 Answer

up vote 3 down vote accepted

Well the gradient is defined as the vector of partial derivatives so that it will exist if and only if all the partials exist. A zero gradient is still a gradient (it's just the zero vector) and we sometimes say that the gradient vanishes in this case (note that vanish and does not exist are different things). A function can still be well defined at a point without the gradient existing, in fact a function can still be continuous without the gradient existing.

share|improve this answer
    
One might define $\nabla f(x) $ to be the unique vector $g$ such that $f(x +h) =f(x) +g^T h +o(h) $ as $|h| \to 0$ (assuming such a $g$ exists). –  littleO Nov 6 '12 at 22:03
    
Thanks! From what I understand tho, if a partial derivative of the gradient does not exist, it means that it is defined as a critical point. –  Yamato C Nov 6 '12 at 22:07
    
@littleO That definition holds only for differentiable functions in which case it just reduces to the normal definition. –  EuYu Nov 6 '12 at 22:11
    
Is this critical point the point where the gradient does not exist? I don't believe so since, from what I have learned, this point is either a local/absolute maximum, minimum or saddle point. –  Yamato C Nov 6 '12 at 22:14
    
A critical point is typically defined as a point in which the gradient vanishes. It'll be a local max/min or saddle depending on the definiteness of the Hessian. –  EuYu Nov 6 '12 at 22:17
show 5 more comments

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.