Activity: The Master Formula (editing)

For an $n-$dimensional function $f$, the gradient of $f$ at a point is an $n-$dimensional vector 

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.

The magnitude of the gradient is the value of the slope in the direction of greatest increase