Answer
a) See the explanation below.
b) See the explanation below.
c) See the explanation below.
Work Step by Step
a) When $f$ is differentiable, then the gradient vector $\nabla f$ for a function $f$ can be calculated as:
$\nabla f(x,y)=\dfrac{\partial f}{\partial x}i+\dfrac{\partial f}{\partial y}j$ (for two variables $(x,y)$)
and
$\nabla f(x,y,z)=\dfrac{\partial f}{\partial x}i+\dfrac{\partial f}{\partial y}j+\dfrac{\partial f}{\partial z}k$ (for three variables $(x,y)$)
b) The directional derivative $D_uf$ in terms of $\nabla f$ can be expressed as:
$D_uf(x,y)=\nabla f(x,y) \cdot u$ (for two variables $(x,y)$)
or, $D_uf(x,y,z)=\nabla f(x,y,z) \cdot u$ (for three variables $(x,y)$)
c) When $f$ is differentiable, then the gradient vector $\nabla f$ for a function $f$ signifies the direction of maximum rate of increase of $f$ and also orthogonal to the level curves of the function $z=f(x,y)$
