Problem 9 - Advanced Engineering Mathematics

Question

Find the Jacobian of the transformation \(T\) from the \(u v\)-plane to the \(x y\)-plane. $$ u=\frac{y}{x^{2}}, v=\frac{y^{2}}{x} $$

Step-by-Step Solution

Answer

The Jacobian of the transformation is the determinant of the Jacobian matrix.

Step 1: Understand the Transformation

The transformation from the \(uv\)-plane to the \(xy\)-plane is given by the equations \(u = \frac{y}{x^2}\) and \(v = \frac{y^2}{x}\). To find the Jacobian, we need to express \(x\) and \(y\) in terms of \(u\) and \(v\).

Step 2: Express x and y in Terms of u and v

From the equation \(u = \frac{y}{x^2}\), we can express \(y = ux^2\). Similarly, from \(v = \frac{y^2}{x}\), substituting \(y = ux^2\) gives \(v = \frac{(ux^2)^2}{x} = u^2 x^3\). Solving these together allows us to express \(x\) and \(y\) in terms of \(u\) and \(v\).

Step 3: Differentiate with Respect to u and v

Find the partial derivatives necessary for the Jacobian matrix: \(\frac{\partial x}{\partial u}\), \(\frac{\partial x}{\partial v}\), \(\frac{\partial y}{\partial u}\), and \(\frac{\partial y}{\partial v}\). This typically involves solving the system from Step 2 to express \(x\) and \(y\). This process might be complex and require implicit differentiation.

Step 4: Form the Jacobian Matrix

The Jacobian matrix \(J\) for the transformation from \(uv\)-plane to \(xy\)-plane is given by: \[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} \]

Step 5: Calculate the Determinant of the Jacobian Matrix

The determinant of the Jacobian matrix represents the Jacobian of the transformation. Calculate the determinant, which is:\[ \text{det}(J) = \left( \frac{\partial x}{\partial u} \cdot \frac{\partial y}{\partial v} \right) - \left( \frac{\partial x}{\partial v} \cdot \frac{\partial y}{\partial u} \right) \]

Key Concepts

Transformation

A transformation is a way of mapping coordinates from one space to another. In the context of the exercise, we are dealing with a transformation from the \(uv\)-plane to the \(xy\)-plane. This means we have equations that link the \((u, v)\) coordinates to new \((x, y)\) coordinates. Transformation is vital in various fields, such as machine learning and 3D graphics.

To perform this transformation, we use the two equations given:

• \(u = \frac{y}{x^2}\)
• \(v = \frac{y^2}{x}\)

Our goal is to express \(x\) and \(y\) in terms of \(u\) and \(v\) and understand how they are interconnected. This transformation provides insights into how one set of coordinate changes affects another set, essential for understanding Jacobean transformations.

Partial Derivatives

Partial derivatives are crucial when dealing with functions of multiple variables. In this exercise, we use partial derivatives to examine how each variable changes when others are held constant. Partial derivatives help us understand the behavior of complex systems by isolating individual variable effects.

For the given transformation, noting how \(x\) and \(y\) vary based on changes in \(u\) and \(v\) involves computing the following partial derivatives:

• \(\frac{\partial x}{\partial u}\): how \(x\) changes when \(u\) changes, keeping \(v\) constant
• \(\frac{\partial x}{\partial v}\): how \(x\) changes when \(v\) changes, keeping \(u\) constant
• \(\frac{\partial y}{\partial u}\): how \(y\) changes when \(u\) changes, keeping \(v\) constant
• \(\frac{\partial y}{\partial v}\): how \(y\) changes when \(v\) changes, keeping \(u\) constant

These derivatives form the building blocks of the Jacobian matrix.

Jacobian Matrix

The Jacobian matrix is a tool that packages all partial derivatives of a transformation neatly. It is a matrix representation that helps to visualize and analyze the relationship between variables in multi-dimensional spaces. This becomes essential in fields such as computer vision and engineering.

For our transformation, the Jacobian matrix \(J\) is constructed as follows: \[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} \]Each element of the matrix is a partial derivative reflecting how one variable changes with respect to another. This matrix is not only valuable for understanding transformations but is also used for solving systems of equations and optimization problems.

Determinant

The determinant of the Jacobian matrix gives us important information about the transformation's behavior. Specifically, it helps determine whether the transformation is invertible and, consequently, if unique solutions exist for the transformations.

In our context, the expression for the determinant is: \[ \text{det}(J) = \left( \frac{\partial x}{\partial u} \cdot \frac{\partial y}{\partial v} \right) - \left( \frac{\partial x}{\partial v} \cdot \frac{\partial y}{\partial u} \right) \]Calculating this determinant helps us understand if the transformation causes stretching or compression, or if the transformation could potentially collapse space dimensions. It offers insights into the stability and feasibility of mapping from \(uv\)-plane to \(xy\)-plane.