Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the transformation \(T: x=g(u, v), y=h(u, v)\) is linear in \(u\) and \(v,\) then the Jacobian is a constant. b. The transformation \(x=a u+b v, y=c u+d v\) generally maps triangular regions to triangular regions. c. The transformation \(x=2 v, y=-2 u\) maps circles to circles.

Short answer

Based on the analysis and solution, the answers to the three statements are: (a) True: A transformation linear in \(u\) and \(v\) results in a constant Jacobian. (b) True: The given linear transformation maps triangular regions to triangular regions. (c) False: The given transformation does not map circles to circles. A counterexample is provided in the solution.

Step-by-step solution

(a) Checking if a linear transformation has a constant Jacobian

The transformation is given by: \(x=g(u,v)\) \(y=h(u,v)\) First, we'll compute the partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v\), and then compute the Jacobian. \(\frac{\partial x}{\partial u} = g_u\) \(\frac{\partial x}{\partial v} = g_v\) \(\frac{\partial y}{\partial u} = h_u\) \(\frac{\partial y}{\partial v} = h_v\) The Jacobian, denoted \(J\), is given by the determinant of the following matrix: \(J = \begin{vmatrix}g_u & g_v \\ h_u & h_v\end{vmatrix} = g_u h_v - g_v h_u\) Since \(g\) and \(h\) are linear in \(u\) and \(v\), their partial derivatives are constant. Therefore, the Jacobian \(J\) is also constant. So, statement (a) is true.

(b) Verifying if the transformation maps triangular regions to triangular regions

The given transformation is: \(x= au+bv\) \(y= cu+dv\) Let's consider three points \((u_1, v_1), (u_2, v_2),\) and \((u_3, v_3)\) that form a triangle in \(uv\) plane. After transformation, we have three points in \(xy\) plane: \((x_1, y_1) = (au_1+bv_1, cu_1+dv_1)\) \((x_2, y_2) = (au_2+bv_2, cu_2+dv_2)\) \((x_3, y_3) = (au_3+bv_3, cu_3+dv_3)\) Let's examine the determinant formed by these three new points: \(\Delta = \begin{vmatrix}x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1\end{vmatrix}\) Expanding this determinant: \(\Delta = \begin{vmatrix}y_2 & 1 \\ y_3 & 1\end{vmatrix} x_1 - \begin{vmatrix}x_2 & 1 \\ x_3 & 1\end{vmatrix} y_1 + \begin{vmatrix}x_2 & y_2 \\ x_3 & y_3\end{vmatrix}\) Using the transformed coordinates, we get: \(\Delta = [(cu_2+dv_2)(cu_3+dv_3) - (cu_3+dv_3)(cu_2+dv_2)](au_1+bv_1) - [\ldots](cu_1+dv_1) + [\ldots]\) Clearly, the determinant is non-zero since it is a linear combination of the three original points, which implies that the transformed points also form a triangle. So, statement (b) is true.

(c) Checking if the transformation maps circles to circles

The given transformation is: \(x=2v\) \(y=-2u\) Suppose we have a circle centered at \((u_0,v_0)\) with radius \(r\). The equation of the circle in the \(uv\) plane is: \((u-u_0)^2 + (v-v_0)^2 = r^2\) Applying the transformation, we have: \(u = -\frac{y}{2}\) \(v = \frac{x}{2}\) Substituting these into the circle equation: \(\left(-\frac{y}{2} - u_0\right)^2 + \left(\frac{x}{2} - v_0\right)^2 = r^2\) Expanding and simplifying: \(\frac{y^2}{4} + 2yu_0 + u_0^2 + \frac{x^2}{4} - 2xv_0 + v_0^2 = r^2\) The transformed equation is of the form \(Ax^2 + By^2 + Cxy + Dx + Ey + F = 0\), which represents an ellipse, not a circle, in general. Therefore, the transformation does not map circles to circles. So, statement (c) is false. A counterexample is the circle \((u-1)^2 + (v-1)^2 = 1\), which after transformation becomes \(\frac{x^2}{4} + \frac{y^2}{4} - x + 2y = 0\), which is an ellipse.

Key concepts

Linear Transformation

A linear transformation is a function that maps vectors from one space to another, where the function is defined by a matrix. This matrix is crucial because it determines how the transformation will alter any vector input. The properties of a linear transformation include:

• Additivity: The transformation of the sum of two vectors equals the sum of the individual transformations of those vectors.
• Homogeneity: The transformation of a scalar multiple of a vector equals the scalar multiple of the transformation of that vector.

When transformations are linear, they preserve linear relationships, like parallelism and the origin remains fixed in the transformation process.
This means, if you have a system represented by linear equations, converting it with a linear transformation will preserve many of the original system's properties.

Determinant

The determinant is a scalar value that can be computed from a square matrix. It provides critical insights into the matrix's properties and the transformations it represents. The determinant is particularly significant because:

• If the determinant is zero, the transformation is singular, meaning it squashes the space it maps to into a lower dimension (e.g., a plane to a line).
• A non-zero determinant implies the transformation is invertible and therefore unique.
• The absolute value of the determinant provides a scale factor. It measures how much the transformation expands or contracts areas or volumes through the linear transformation.

In the context of transformations using Jacobians, the determinant helps establish if areas are preserved in size during transformations.

Coordinate Transformation

Coordinate transformations are used to change from one coordinate system to another. These transformations are essential in simplifying complex problems by switching to a coordinate system where the problem is easier to solve.
Key concepts when dealing with coordinate transformations include:

• Preserving geometric properties such as shapes and distances as much as possible.
• Altering the basis of the space from one coordinate set to another, such as from Cartesian coordinates to polar coordinates.
• Utilizing Jacobians to adjust volume elements to the new coordinate system.

Coordinate transformation can change the layout and scale of geometric figures. It provides tools to examine how transformations affect geometry, like transforming a circular region in one system to an elliptical pattern in another.