Question

Find the Jacobians \(\partial(x, y) / \partial(u, v)\) of the given transformations from variables \(x, y\) to variables \(u, v:\). $$\begin{aligned} &x=\frac{1}{2}\left(u^{2}-v^{2}\right)\\\ &y=u v \end{aligned}$$ \((u \text { and } v\) are called parabolic cylinder coordinates).

Short answer

\(u^2 + v^2\)

Step-by-step solution

- Write Down the Transformations

Start by noting the given transformations: \[x = \frac{1}{2}(u^2 - v^2)\]\[y = uv\]

- Compute Partial Derivatives of x with Respect to u and v

To find the Jacobian, compute \(\frac{\partial x}{\partial u}\) and \(\frac{\partial x}{\partial v}\): \[\frac{\partial x}{\partial u} = \frac{1}{2} \cdot 2u = u\]\[\frac{\partial x}{\partial v} = \frac{1}{2} \cdot (-2v) = -v\]

- Compute Partial Derivatives of y with Respect to u and v

Next, compute \(\frac{\partial y}{\partial u}\) and \(\frac{\partial y}{\partial v}\): \[\frac{\partial y}{\partial u} = v\]\[\frac{\partial y}{\partial v} = u\]

- Form the Jacobian Matrix

Construct the Jacobian matrix \(J\) with these partial derivatives: \[J = \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} = \begin{pmatrix} u & -v \ v & u \end{pmatrix}\]

- Compute the Determinant of the Jacobian Matrix

Finally, calculate the determinant of the Jacobian matrix to find the Jacobian: \[\text{det}(J) = \begin{vmatrix} u & -v \ v & u \end{vmatrix} = u \cdot u - (-v) \cdot v = u^2 + v^2\]

Key concepts

partial derivatives

Partial derivatives are essential in understanding how a function changes as its variables change.
Unlike ordinary derivatives, which deal with functions of a single variable, partial derivatives apply to functions of multiple variables.
Consider the function given in the exercise:
\(x = \frac{1}{2}(u^2 - v^2)\) and \(y = uv\).
To find how these functions change with respect to each variable, we compute their partial derivatives. For instance:
\( \frac{\text{Partial derivative of } x \text{ w.r.t. } u}{\text{Partial derivative of } x \text{ w.r.t. } v}\)
In this case, partial derivatives help us form the Jacobian matrix, which is key for transformations between coordinate systems.

parabolic cylinder coordinates

Parabolic cylinder coordinates \( (u, v) \) are a way to describe points in a plane, much like Cartesian coordinates (\$x,y\$) or polar coordinates (\$r, \theta\$).
They are particularly useful in problems with parabolic symmetry.
In our exercise:
\(x = \frac{1}{2}(u^2 - v^2)\)
and \(y = uv\).
These coordinates can simplify the equations of parabolas and other curves that are otherwise complex in Cartesian coordinates.
When solving problems in parabolic cylinder coordinates, it's crucial to understand how these coordinates relate to each other, which is where the Jacobian comes into play to make sense of the transformations.

transformation matrix

The transformation matrix, or Jacobian matrix, is a tool that describes how one set of coordinates maps to another.
For transformations between coordinates, each element of the matrix represents a partial derivative of one variable with respect to another.
In our problem:
\( J = \begin{pmatrix} \frac{\text{Partial derivative of }x\text{ w.r.t. }u} & \frac{\text{Partial derivative of }x \text{ w.r.t. } v}\frac{\text{Partial derivative of } y\text{ w.r.t. } u} & \frac{\text{Partial derivative of } y \text{ w.r.t. } v} \end{pmatrix} = \begin{pmatrix} u & -v \ v & u \end{pmatrix}\)
This matrix helps us see how the changes in one set of variables (\$u, v\$) affect changes in the other set (\$x, y\$). The determinant of this matrix gauges the 'stretch' or 'compression' of transformations.
In simpler terms, it tells us how the areas change as we map from one set of coordinates to another. In our case, \( \text{det}(J) = u^2 + v^2 \).