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

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.

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.

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Partial Derivative

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