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Gradient Covariance
More Tech Stuff:
Indexing Books: Lessons in Language Computations
Constructing a Mandelbrot Set Based Logo with Visual Basic.NET and Fireworks
page 2 of 4
The gradient of (∇f) is a vector with the same number of dimensions as the coordinate system used. It points in the direction of greatest change in the scalar field.
To specify the components of the vector ∇f we can use subscripts. For the dimensions in general we can use the subscript i, (and we will also use j when we need more than one subscript, which we will). Thus the third component of ∇f would be 
, in general we would write 
and
For all that, our example is in just two dimensions, 
and 
.
Our field of scalar values will be created by the function
This function, involving squaring and then halving each component, was chosen because it makes for nice partial derivatives [Need help? See taking partial derivatives]. We get F’s gradient:
And so
All this was assuming the standard Cartesian coordinate system. Now let’s see what happens if we change systems.
See also: gradient at Wikipedia; gradient at Wolfram
The New Coordinate System
To distinguish the new coordinate system from the 
already established, we call its axes the 
, the barred coordinate system. In this sort of exercise it is very important to keep track of the 
’s versus the ’s.
The simplest coordinate changes are no different that measuring in the English system (inches), then converting into the metric system (centimeters). Then in one dimension the change can be expressed as:
Most coordinate system changes not simple scale changes from one Cartesian system to another. Even in these more complicated transformations we can (usually) express each in terms of the set of ’s. In our particular example we will use a new coordinate system expressed in a modestly complicated way from our original coordinate system by:
continued Gradient Covariance page 3