What is the definition of a gradient?

Question

It has been a while since I have done any vector calculus,

is this statement true?

$\nabla f(x,y,z) = 0 \iff \dfrac{\partial f}{\partial x} + \dfrac{\partial f}{\partial y} + \dfrac{\partial f}{\partial z} = 0$

or is it only when the partials are equal to zero on their own individually?

Answer 1

The gradient is a vector of partial derivatives, not a sum of partial derivatives. A vector is zero if and only if each of its components is zero.

Our, as TravisJ put it,

The gradient is zero when each component of the gradient is zero (since the gradient is a vector). The partial derivatives are the components of the vector, so you need every partial derivative to be zero in order for the gradient to be zero.