Given a function $f(x,y)$, its gradient is defined to be: $\nabla f(x,y) = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j}$.
[$\hat{i}$ and $\hat{j}$ are unit vectors in the $x$ and $y$ direction]
Given this definition, the gradient vector will always be parallel to the $x$-$y$ plane.
The gradient is also supposed to be perpendicular to the tangent of a plane (its
"normal" vector).
How, however, could it be perpendicular to the tangent of the plane if it is always parallel to the $x$-$y$ plane?
Best Answer
This isn't true. The gradient vector is perpendicular to the curve $f(x,y)=0$, not perpendicular to the plane containing the curve.