Problem 17 - Calculus Early Transcendentals

Question

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=3 u, y=-3 v$$

Step-by-Step Solution

Answer

Question: Determine the Jacobian \(J(u,v)\) of the transformation \(T(u,v) = (3u, -3v)\). Answer: The Jacobian \(J(u,v)\) of the given transformation is -9.

Step 1: Determine the partial derivatives

Compute the partial derivatives of \(x\) and \(y\) with respect to \(u\) and \(v\): $$\frac{\partial x}{\partial u} = \frac{\partial (3u)}{\partial u} = 3$$ $$\frac{\partial x}{\partial v} = \frac{\partial (3u)}{\partial v} = 0$$ $$\frac{\partial y}{\partial u} = \frac{\partial (-3v)}{\partial u} = 0$$ $$\frac{\partial y}{\partial v} = \frac{\partial (-3v)}{\partial v} = -3$$

Step 2: Construct the Jacobian matrix

Now that we have all the partial derivatives, we can write down the Jacobian matrix: $$J(u, v) = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} 3 & 0 \\ 0 & -3 \end{bmatrix}$$

Step 3: Compute the determinant of the Jacobian matrix

Compute the determinant of the Jacobian matrix: $$\det(J(u, v)) = (3 \times -3) - (0 \times 0) = -9$$ So the Jacobian \(J(u, v)\) for the given transformation is: $$J(u, v) = -9$$

Key Concepts

Partial Derivatives

When dealing with functions that depend on several variables, it is often essential to understand how changes in one variable affect changes in the function's value. This is where partial derivatives come into play. A partial derivative is the derivative of a multivariable function with respect to one variable, keeping all other variables constant. This concept is quite similar to a regular derivative but applied to functions with more than one variable.
In the given transformation, we have the variables expressed as:

• \(x = 3u\)
• \(y = -3v\)

To find the Jacobian, we need to calculate the partial derivatives \(\frac{\partial x}{\partial u}, \frac{\partial x}{\partial v}, \frac{\partial y}{\partial u},\) and \(\frac{\partial y}{\partial v}\). These partial derivatives give us how the transformation scales or rotates the inputs \(u\) and \(v\) into \(x\) and \(y\). They provide insight into how sensitive the function is to a change in one specific direction, while keeping everything else unchanged.
Understanding partial derivatives is crucial because they form the components of the Jacobian matrix, which we discuss next.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix. In the context of Jacobians, the determinant is crucial because it measures how the area or volume scales under a transformation. In other words, it tells us how much the transformation will stretch or compress a small region around a point.
For our transformation, the Jacobian matrix is:\[J(u, v) = \begin{bmatrix}3 & 0 \0 & -3\end{bmatrix}\]The determinant of this Jacobian matrix is calculated as:\[\det(J(u, v)) = (3 \times -3) - (0 \times 0) = -9\]This result means that the transformation inversely scales the area by a factor of 9. A negative determinant indicates that the transformation also involves a reflection, flipping the orientation of the transformed object. Understanding the determinant helps in analyzing how functions behave under transformation, which is crucial for coordinate transformation topics.

Coordinate Transformation

Coordinate transformation is a critical concept in mathematics and engineering, which involves changing the coordinate system to simplify problems or to better understand the behavior of a system. A transformation, like the one given in our problem, maps a set of variables \((u, v)\) to a new system \((x, y)\) using specific rules or equations.
This exercise demonstrates a linear transformation where \(x = 3u\) and \(y = -3v\). Such transformations are often used in physics, computer graphics, and robotics to move between different coordinate systems for ease of calculation or to gain new viewpoints on a problem.
By understanding the Jacobian matrix and its determinant, we obtain valuable information about how the transformation affects areas, volumes, and orientations in space. Coordinate transformations in vector spaces are vital for integrating, differentiating, and solving equations in more complex systems. Whether you're rotating a space in 3D graphics or moving between inertial frames of reference in physics, mastering this concept broadens your understanding of how different systems relate to each other.