Concept: *The Magnitude of the Gradient (editing)

Prerequisite concepts
The Master Formula
The gradient is a vector

For an $n-$dimensional function $f$, the gradient of $f$ at a point is an $n-$dimensional vector 
*The Direction of the Gradient
*The Magnitude of the Gradient
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*The Magnitude of the Derivative

FIXME
The Master Formula

*The Magnitude of the Gradient

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

Representations used

The Hot Plate

The magnitude of the gradient is proportional to the density of contour lines

Contour Maps

As adjacent contour lines represent equal changes in the function, the denser the contour lines, the quicker the function is changing. Therefore, the magnitude of the gradient is greater where the contour lines are denser.
There is a partial derivative in every direction at any point
There is a tangent line in every direction at every point
The value of a partial derivative depends on the value(s) of what is held constant
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NEEDS NAME

$\vec \nabla f$

ENTER LONG DESCRIPTION
The Hill
The magnitude of the gradient is proportional to the density of contour lines
The direction of the gradient is perpendicular to contour lines
The gradient is a local quantity
The gradient lives in the domain and has the same number of spatial dimensions as the original function
Visualizing Gradient
The gradient lives in the domain and has the same number of spatial dimensions as the original function
The gradient is a local quantity
The direction of the gradient is perpendicular to contour lines