Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

When you want to locate the location for the maximum rate of change of a single variable function, then one method is to use the second derivative test(f''=0).

In the multivariable case we can use the gradient. I know that the gradient vector points in the direction of where the function's rate of change is maximum but I was unable to solve my problem unless I was given the hint that I should check maximum of $\left | \bigtriangledown f \right |^{2}$. I was asked to determine at what location the steepest point is. I cannot find an argument in my textbook about this so I ask here: What does $\left | \bigtriangledown f \right |^{2}$ describe?

share|cite|improve this question
up vote 0 down vote accepted

$|\nabla f|^2$ is the squared magnitude of the gradient. The gradient vector describes the direction of greatest increase at any given point. Finding the maximum magnitude of this vector tells you where the increase is greatest (or more precisely, greatest in magnitude).

share|cite|improve this answer
Well since $\left | \triangledown f \right |$ describes the direction of the greatest increase, then I should maximize $\left | \triangledown f \right |$? Is not that right? Why squared? – EricAm Jan 29 '13 at 20:30
$\nabla f$ is a vector, and so it describes the direction of greatest increase. $|\nabla f|$ is that vector's magnitude, defined to be positive, so there is no difference between maximizing it and maximizing its square. – Muphrid Jan 29 '13 at 20:39

Your Answer


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.