91Ó°ÊÓ

The Jacobian matrix is an important tool in calculus, especially when dealing with transformations between coordinate systems. It consists of all possible first-order partial derivatives of a vector function. In simpler terms, when you have a transformation from variables \(u, v\) to \(x, y\), the Jacobian matrix helps describe how small changes in \(u, v\) affect \(x, y\).

For our problem, the Jacobian matrix is a 2x2 matrix that looks like this:

• First row: \(\frac{\partial x}{\partial u}, \frac{\partial x}{\partial v}\)
• Second row: \(\frac{\partial y}{\partial u}, \frac{\partial y}{\partial v}\)

The elements in this matrix are partial derivatives that measure how \(x\) and \(y\) change with respect to \(u\) and \(v\). Calculating these derivatives gives us the insight needed to determine where the transformation fails to be one-to-one.

Partial derivatives are essential when working with functions of multiple variables. They measure how much a function changes as one specific variable changes, while keeping all other variables constant.

In the exercise, for example, to calculate \(\frac{\partial x}{\partial u}\), you consider changes in \(x\) as \(u\) changes, while \(v\) remains fixed. This results in a formula that captures the local rate of change:

• \(\frac{\partial x}{\partial u} = 2u\)
• \(\frac{\partial x}{\partial v} = v^3\)
• \(\frac{\partial y}{\partial u} = 1\)
• \(\frac{\partial y}{\partial v} = \sin v\)

These partial derivatives form the components of our Jacobian matrix, allowing us to further analyze the transformation's behavior.

In mathematics, a one-to-one transformation (injective) means that each element in the domain maps to a unique element in the range. The implication is that no two different input values produce the same output.

Checking for one-to-oneness involves finding where the Jacobian determinant equals zero. When this happens, it signifies critical points where the transformation possibly loses its one-to-one property.

For the given transformation, the deterrent to being one-to-one is at points where:

• \(2u \sin v = v^3\)

By solving this equation, we identify points such as \(u = 0\) and \(v = n\pi\) (where \ is an integer) where the mapping is not one-to-one.

Graphing transformations helps visualize how points in the input plane map to the output plane. When examining transformations, it's crucial to highlight regions where the transformation loses its one-to-one nature, as these are pivotal points of interest.

From our solutions, graphing involves plotting all points \(u, v\) satisfying the conditions:

• Vertical line at \(u=0\), extending from \(-\pi \leq v \leq \pi\)
• Horizontal lines at \(v=0\) and other integer multiples of \(\pi\) within the range for \(u\)

These lines show the exact locations where the transformation \(T\) is not one-to-one on the \(uv\) plane, helping you understand and interpret the behavior of the transformation.