Directional derivatives

Notes:


Suppose we have a vector $\vec{v} =\lt v_1, v_2\gt$. The unit vector will be:

$\vec{v} = \frac{1}{\sqrt{v^2_1 + v^2_2}} \lt v_1,v_2\gt$

Suppose we have a vector $\vec{v} =\lt v_1, v_2,v_3\gt$. The unit vector will be:

$\vec{v} = \frac{1}{ \sqrt{v_1^2 + v_2^2 + v_3^2} } <v_1,v_2,v_3>$

When given an angle of a direction ($\theta$ ), we say that the unit vector (that points to the direction) is:

$\vec{u} = \lt \cos \theta, \sin \theta \gt$

Directional Derivatives of 2 Variable Functions
A Directional Derivative is the rate of change (of $x$ and $y$) of a function at a point $P=(x_0,y_0,z_0)$, at the direction of the unit vector

Suppose there is a 2-variable function $z=f(x,y)$. Then the directional derivative is:

$D_{\vec{u}}f(x,y) = f_x(x,y)a + f_y(x,y)b$

where the $\vec{u} = <a,b>$ is the unit vector that points in the direction of change. Directional Derivatives of 3 Variable Functions
Suppose there is a 3-variable function $w=f(x,y,z)$. Then the directional derivative is:

$D_{\vec{u}}f(x,y,z) = f_x(x,y,z)a + f_y(x,y,z)b + f_z(x,y,z)c$

where the $\vec{u} = <a,b,c>$ is the unit vector that points in the direction of change.