Verify (7.16) in three ways: (a) Differentiate equations...

Verify $(7.16)$ in three ways:
(a) Differentiate equations (7.6).(b) Take differentials of $(7.5)$ and solve for $d r$ and $d \theta$,
(c) Find $A^{-1}$ in $(7.15)$ from $A$ in $(7.13)$; note that this is (b) in matrix notation.

Key Concepts

Total Differentials

Total differentials capture how a function changes in response to changes in all of its independent variables. This concept is crucial when one needs to determine the small change in a function that depends on multiple variables. In coordinate transformations, taking the total differential of an expression helps unravel how each coordinate contributes to the overall change, thereby underpinning the process of solving for specific differential elements.

Jacobian Matrix and Matrix Inversion

The Jacobian matrix organizes all of the first-order partial derivatives of a set of functions that define a transformation. It encapsulates the local linear approximation of the transformation and is essential for converting between differential forms. Matrix inversion within this context is used to reverse the transformation, allowing one to express the differentials of the original variables in terms of the new ones. This step is especially important in systems of equations where the relationship between variables is expressed in matrix form.

Differentiation of Transformation Equations

This concept involves taking the derivative of equations that relate two sets of coordinates. By differentiating these transformation equations, one obtains relationships between the rates of change (or differentials) of one set of variables in terms of the other. It is a fundamental method in calculus used to link variables through their small changes, particularly in contexts such as transforming between Cartesian and polar coordinates.

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