Gradient vectors

The gradient vector (denoted as $\nabla f$) is a vector where all the components are partial derivatives of the function in respect to each variable. Also known as the direction with the greatest increase of $f$

For example, consider the function $f(x,y,z)$. Then,

$\nabla f = \lt f_x, f_y, f_z\gt$

If you want to find the gradient of a specific point $(x_0, y_0, z_0)$, then

$\nabla f(x_0, y_0, z_0)= \lt f_x(x_0, y_0, z_0),f_y(x_0, y_0, z_0), f_z(x_0, y_0, z_0)\gt$

Gradients are useful for finding the tangent plane.

Recall that the equation of a plane is:

$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$

The gradient vector is actually the normal vector that is orthogonal to the tangent plane at $(x_0, y_0, z_0)$. So that means:

$a=f_x(x_0, y_0, z_0)$
$b=f_y(x_0, y_0, z_0)$
$c=f_z(x_0, y_0, z_0)$

There are times in which instead of finding the normal vector, we want the normal line. Recall that the formula for a vector equation is:

$r(t)= \lt x_0, y_0, z_0\gt+ t\lt a,b,c\gt$

Since the gradient is the direction of the vector, and we already have an initial point $(x_0, y_0, z_0)$, then the normal line is:

$r(t)= \lt x_0, y_0, z_0\gt + t \nabla f(x_0, y_0, z_0)$