The tree diagram (some people also call it a dependency graph) is an alternative visual tool to help construct chain rule.
In a tree diagram, the dependent variable sit on top, and independent variables sit on bottom, with intermediate variables sit in the middle.
Example
For example, to construct chain rule for a function where each is depending on two variables and , we can have the following diagram:
Chain Rule and Coordinate Transformation
One application of the chain rule is to transform the coordinate system. When a problem has circular symmetry for instance, it doesn’t make sense to write in Cartesian coordinates; the problem is much easier to solve in polar coordinates.
Find the gradient in polar form. Then express it in unit vectors of polar coordinate ()
Solution: By the chain rule we have
We also know that and , and
Then
and
So we have
We can similarly get
Putting together,
Related
total differential - A lot of problems can be solved by either chain rule or expanding the total differential, though using the chain rule is usually the easier way