====== Gradient ======

The //gradient// of a multi-variable function gives a vector pointing in the direction of steepest slope from any point.
The elements of the vector are the [[partial derivative]]s of the function at that point.  
The //gradient// is often represented by the nabla symbol $\nabla$.

For a two-variable function $f(x,y)$, the gradient is the 2D vector given by:
$$\nabla f(x,y) = \begin{bmatrix}
\frac{\partial }{\partial x}f(x,y) & 
\frac{\partial }{\partial y}f(x,y)
\end{bmatrix}$$

==== Example 1. ====
{{ ::gradient_example1.png|Gradient Example 1}}
For the function

$$f(x,y) = 8-(x^2 + y^2)$$

the gradient is

$$\nabla f(x,y) = \begin{bmatrix}
\frac{\partial }{\partial x}f(x,y) & 
\frac{\partial }{\partial y}f(x,y)
\end{bmatrix} \\
= \begin{bmatrix} -2x & -2y \end{bmatrix}$$

The graph of the function is a 3D surface shaped as a convex parabola, an upside-down bowl.  
A contour map lies underneath the graph at $z=0$.

For the random point $x=.75, y=-1.5, z=5.18750$,

the gradient is given as
$$ \nabla f(.75,-1.5) = \begin{bmatrix}-1.5 & 3.0\end{bmatrix} $$

shown as the red line segment lying across the contour map, perpendicular to the contour lines.

==== Example 2.====

{{ ::gradient_example2.png|Gradient Example 2}}
For the function

$$f(x,y) = (x^2 + y^2)$$

the gradient is

$$\nabla f(x,y) = \begin{bmatrix}
\frac{\partial }{\partial x}f(x,y) & 
\frac{\partial }{\partial y}f(x,y)
\end{bmatrix} \\
= \begin{bmatrix} 2x & 2y \end{bmatrix}$$

The graph of the function is a bowl-shaped concave parabola.

For the random point at $x=.75, y=-1.5, z=2.8125$,

the gradient is given as
$$ \nabla f(.75,-1.5) = \begin{bmatrix}1.5 & -3.0\end{bmatrix} $$

shown as the red line segment on the contour map, pointing away from center.

==== Example 3. ====

{{ ::gradient_example3.png|Gradient Example 3}}
This example is not working yet.

~CLEAR~

==== See Also ====

[[Partial Derivative]]
==== References ====

The octave scripts for these examples are located here: http://samantha.voyc.com/doc/script/octave/