PX153 - H1 - directional derivative and gradient vector of scalar functions

consider a continuous differentiable function, $f (x, y, z)$ . we want to estimate the change of $f$ between two points in space
let $\vec{a} = (x_{a}, y_{a}, z_{a})$ be the position vector to point $A$
let $B$ be a vector $\vec{u}$ from $A$
any point on the line from $A$ to $B$ can be written as $\vec{r} = \vec{a} + t \cdot \vec{u}$ , for $t \in [0, 1]$
we can use a taylor expansion to evaluate the values of $f$ close to $A$ :

f (\vec{r}) = f (\vec{a}) + \frac{\partial f}{\partial \vec{r}} |_{a} (\vec{r} - \vec{a}) + . . .

f (x, y, z) = f (x_{a}, y_{a}, z_{a}) + \frac{\partial f}{\partial x} (x - x_{a}) + \frac{\partial f}{\partial y} (y - y_{a}) + \frac{\partial f}{\partial z} (z - z_{a})

$\vec{r} = \vec{a} + t \cdot \vec{u}$
$⟹ x = x_{a} + t u_{x}$
$y = y_{a} + t u_{y}$
$z = z_{a} + t u_{z}$

Δ f = \frac{\partial f}{\partial x} t u_{x} + \frac{\partial f}{\partial y} t u_{y} + \frac{\partial f}{\partial z} t u_{z}

we can write this as a scalar product:

Δ f = (\frac{\partial f}{\partial x} \hat{\vec{i}} + \frac{\partial f}{\partial y} \hat{\vec{j}} + \frac{\partial f}{\partial z} \hat{\vec{k}}) |_{\vec{a}} \cdot t \vec{u}

we this define the gradient operator (vector):

\vec{\nabla} f = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} + \frac{\partial f}{\partial z} \hat{k} = (\hat{i} \frac{\partial}{\partial x} + \hat{j} \frac{\partial}{\partial y} + \hat{k} \frac{\partial}{\partial z}) f

and we find that:

Δ f = \vec{\nabla} f \cdot (t \vec{u})

the directional derivative is defined as:

D_{u} f = lim_{t \to 0} \frac{f (\vec{r}) - f (s)}{t} = lim_{t \to 0} \frac{Δ f}{t} = \vec{\nabla} f \cdot \vec{u}

this gives the rate of change of function in the direction of $\vec{u}$
- magnitude of $\vec{u}$ should use $\hat{\vec{u}}$