Show that if \(v\) is a vector and \(p\) is a covector at some point of a manifold, then \(p_{a} v_{a}\) is independent of the coordinate system in which it is evaluated.

In the realm of analytical dynamics, vectors and covectors are fundamental entities. Vectors represent quantities that have both magnitude and direction, while covectors, also known as one-forms, are linear functions that map vectors to the real numbers, providing a scalar. An easy way to distinguish them in the language of components is that vectors have upper indices (e.g., \(v^{a}\)), and covectors have lower indices (e.g., \(p_{a}\)).

When dealing with vectors and covectors, it's important to understand their behavior under coordinate transformations. As shown in the exercise, the components of vectors and covectors transform differently. This characteristic ensures that the product \(p_{a} v^{a}\), known as the inner product, is invariant under a change of coordinates — a concept that is foundational in physics and mathematics, where measurements and calculations must have the same value regardless of the coordinate system used.

The concept of coordinate transformation is a critical part of understanding how vectors and covectors behave in different reference frames. In essence, it's the switch from one set of coordinates to another, a process that is neatly described mathematically using transformation matrices. Imagine you're in a city and you switch from using a street map with one set of street names to using another map with a different naming system — that's coordinate transformation.

Mathematically, when the exercise refers to \(x'^{a} = J^{a}_{\phantom{a}b} x^{b}\), it is describing how each new coordinate \(x'^{a}\) is a function of all the old coordinates \(x^{b}\). The Jacobian matrix \(J^{a}_{\phantom{a}b}\) is the 'rule book' for this transformation, telling us how to get from the old coordinates to the new ones. It is essential for ensuring that physical laws remain valid no matter what coordinate system is being used, paving the way for a consistent description of natural phenomena.

At the heart of understanding coordinate transformation lies the Jacobian matrix. The Jacobian matrix is a matrix of all first-order partial derivatives of a vector-valued function with respect to its variables. It can be seen as the 'engine' driving the transformation of vector and covector components when coordinates are changed.

The exercise showcases that a vector transforms with the inverse of the Jacobian matrix \(J^{-1}_{\phantom{a}b}^{\phantom{-1}a}\), while a covector uses the Jacobian matrix itself \(J^{a}_{\phantom{a}b}\). The interplay between the Jacobian and its inverse ensures the inner product remains constant, \(p'_{a}v'^{a} = p_{a}v^{a}\), proving that physical quantities like energy or momentum are preserved across different views of the same physical situation. The elegance of the Jacobian matrix lies in its ability to adapt and apply these transformative rules seamlessly across multiple dimensions and complex systems.