Problem 21 - Thomas Calculus

Question

Find the Jacobian \(\partial(x, y) / \partial(u, v)\) of the transformation \begin{equation} \begin{array}{l}{\text { a. } x=u \cos v, \quad y=u \sin v} \\ {\text { b. } x=u \sin v, \quad y=u \cos v.}\end{array} \end{equation}

Step-by-Step Solution

Answer

For Part (a), the Jacobian is \( u \). For Part (b), the Jacobian is \( -u \).

Step 1: Understand the Concept of the Jacobian

The Jacobian matrix is used to describe the rate of change of a multivariable function. It consists of first-order partial derivatives of the function components.

Step 2: Identify the Functions

For Part (a) of the problem, the functions are given as \( x = u \cos v \) and \( y = u \sin v \). We need to express \( x \) and \( y \) in terms of \( u \) and \( v \). For Part (b), the functions are \( x = u \sin v \) and \( y = u \cos v \).

Step 3: Compute Partial Derivatives for Part a

For Part (a), compute the partial derivatives:- \( \frac{\partial x}{\partial u} = \cos v \)- \( \frac{\partial x}{\partial v} = -u \sin v \)- \( \frac{\partial y}{\partial u} = \sin v \)- \( \frac{\partial y}{\partial v} = u \cos v \)

Step 4: Form the Jacobian Matrix for Part a

The Jacobian matrix for Part (a) is:\[\begin{bmatrix}\cos v & -u \sin v \\sin v & u \cos v\end{bmatrix}\]

Step 5: Calculate the Determinant for Part a

The determinant of the Jacobian matrix for Part (a) is calculated as follows:\[ J = \cos v \cdot (u \cos v) - (-u \sin v) \cdot \sin v \] which simplifies to:\[ J = u \cos^2 v + u \sin^2 v = u(\cos^2 v + \sin^2 v) = u\]

Step 6: Compute Partial Derivatives for Part b

For Part (b), compute the partial derivatives:- \( \frac{\partial x}{\partial u} = \sin v \)- \( \frac{\partial x}{\partial v} = u \cos v \)- \( \frac{\partial y}{\partial u} = \cos v \)- \( \frac{\partial y}{\partial v} = -u \sin v \)

Step 7: Form the Jacobian Matrix for Part b

The Jacobian matrix for Part (b) is:\[\begin{bmatrix}\sin v & u \cos v \\cos v & -u \sin v\end{bmatrix}\]

Step 8: Calculate the Determinant for Part b

The determinant of the Jacobian matrix for Part (b) is calculated as follows:\[J = \sin v \cdot (-u \sin v) - (u \cos v) \cdot \cos v\] which simplifies to:\[ J = -u \sin^2 v - u \cos^2 v = -u(\sin^2 v + \cos^2 v) = -u\]

Step 9: Conclusion

For Part (a), the Jacobian \( \partial(x, y) / \partial(u, v) \) is \( u \). For Part (b), the Jacobian \( \partial(x, y) / \partial(u, v) \) is \( -u \).

Key Concepts

Partial Derivatives

Partial derivatives are vital in understanding how a function changes with respect to each of its variables. Imagine a function with multiple variables, like a landscape. Each variable can be thought of as a direction in which you hike. The partial derivative shows the steepness of the landscape in that specific direction.
When dealing with transformations like those in our exercise, we compute partial derivatives to understand how each transformation component, like \(x\) or \(y\), changes with respect to \(u\) and \(v\).
For instance, if \(x = u \cos v\), computing \(\frac{\partial x}{\partial u}\) gives us \(\cos v\), indicating how \(x\) changes as \(u\) varies while keeping \(v\) constant. Similarly, \(\frac{\partial x}{\partial v}\) equals \(-u \sin v\), showing how \(x\) changes as \(v\) changes.

Determinant Calculation

The determinant of the Jacobian matrix offers significant insights into the nature of the transformation. In simple terms, it tells us how much the transformation scales or changes areas/volumes in a space.
The determinant is calculated from a 2x2 matrix using the formula:

• Multiply the top left and bottom right entries of the matrix.
• Subtract the product of the top right and bottom left entries.

For a transformation, if the determinant equals zero, it means that the transformation squashes the area to a line or point.
In our exercise, we find that for Part (a), the determinant is \(u\), indicating a scaling factor. Meanwhile, for Part (b), the determinant is \(-u\), which flips the area, indicating a reflection and scaling.

Multivariable Functions

Multivariable functions extend the classic functions with a single variable into functions with two or more variables. Think of these as having multiple inputs affecting the output, similar to how ingredients in a recipe combine to make a dish.
These functions are a core aspect of calculus and appear frequently when modeling real-world phenomena like physics, engineering, and economics, where several factors influence outcomes.
The Jacobian matrix is intrinsically linked to these functions, providing a snapshot of how the function changes across all variables. Understanding these functions is crucial in fields requiring optimization or solving systems of equations.
Considering the functions \(x = u \cos v\) or \(y = u \sin v\) in our problem helps us visualize how changes in \(u\) or \(v\) shape the entire transformation, making the Jacobian matrix and its determinant essential tools.