Prerequisite concepts
The gradient is a vector
For an $n-$dimensional function $f$, the gradient of $f$ at a point is an $n-$dimensional vector
Representations used
The components of the gradient are partial derivatives
The direction of the gradient is perpendicular to contour lines
The direction of the gradient shows what way is 'uphill'
The magnitude of the gradient is proportional to the density of contour lines
FIXME
*The Magnitude of the Gradient
*The Direction of the Gradient
The gradient is a vector
The magnitude of the gradient is the value of the slope in the direction of greatest increase
The Master Formula states that a small change in a function $df$ is the dot product of the gradient of the function with a small step $d \vec r$ through the domain of the function: \[df = \vec \nabla f \cdot d \vec r.\] In order to maximize the change in $f$ one must maximize this dot product, which happens when the small step $d \vec r$ is parallel to the gradient. Or, turning this statement around, the gradient points in the direction in which the function is increasing the most.