 |
|
|
|
|
|
|
|
 |
|
||||||
 |
 |
 |
 |
 |
 |
 |
|
Gradient Covariance
More Tech Stuff:
Indexing Books: Lessons in Language Computations
Constructing a Mandelbrot Set Based Logo with Visual Basic.NET and Fireworks
page 4 of 4
Covariance
For a vector acting as a first-order covariant tensor
Here I have introduced the Einstein summation convention. When you see the same subscript two times on the same side of the equation, it means we sum over the range of the subscript, i, or j, or whatever. This may take some getting used to. In this case for each i we get one equation. On the right hand side for those equations we sum over j, as below.
We could have simply derived the barred gradient formula using this formula. Here the formula will be used to check to see if our methods worked. We will need some partial derivatives from the coordinate transformation equations:
Now we are down to algebra, being careful to keep the barred and unbarred coordinates, and the i’s and j’s, straight:
Recall 
, so
So the equation holds. Now for i = 2:
In our specific example, a function generates a set of scalars. The gradient is taken on the scalars. Transforming the coordinates, but leaving the reality of the scalar values in place, means we also get a different formula for the gradient. However, the gradient formulas transform according to the standard rule for covariance.
Aside from prepping for more difficult problems involving gradient transforms or covariance, the reader should note that coordinate transformations can be used to simplify problems. We could have started with the more complex formula for the gradient. Our inverse coordinate transformation then would allow us to change to the simpler formula.
Sources:
Tensor Calculus by David C. Kay. [Schaums Outline of Tensor Calculus
Introduction to Tensor Calculus, Relativity and Cosmology
Thanks to Alin Dumitrescu for noticing transciption errors in the first posting of this web article.
©2011 William P. Meyers