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Chapter 4: Problem 25

In the following exercises, find the Jacobian \(J\) of the transformation.\(x=e^{2 u-v}, y=e^{u+v}\)

Short Answer

Expert verified
The Jacobian determinant \(J\) is \(3e^{3u}\).

Step by step solution

01

Understanding the Transformation

The given transformation is:\[ x = e^{2u-v}, \quad y = e^{u+v} \]We need to find the Jacobian matrix of this transformation with respect to variables \(u\) and \(v\).
02

Compute Partial Derivatives

Calculate the partial derivatives required to form the Jacobian matrix. We need:\[ \frac{\partial x}{\partial u}, \frac{\partial x}{\partial v}, \frac{\partial y}{\partial u}, \frac{\partial y}{\partial v} \]Calculate for \(x\):\[ \frac{\partial x}{\partial u} = \frac{\partial}{\partial u}(e^{2u-v}) = 2e^{2u-v} \]\[ \frac{\partial x}{\partial v} = \frac{\partial}{\partial v}(e^{2u-v}) = -e^{2u-v} \]Calculate for \(y\):\[ \frac{\partial y}{\partial u} = \frac{\partial}{\partial u}(e^{u+v}) = e^{u+v} \]\[ \frac{\partial y}{\partial v} = \frac{\partial}{\partial v}(e^{u+v}) = e^{u+v} \]
03

Form the Jacobian Matrix

Construct the Jacobian matrix \(J\) using the partial derivatives calculated:\[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} 2e^{2u-v} & -e^{2u-v} \ e^{u+v} & e^{u+v} \end{bmatrix} \]
04

Calculate the Determinant

The Jacobian determinant \(J\) is found by evaluating the determinant of the Jacobian matrix:\[ \text{det}(J) = \left(2e^{2u-v} \cdot e^{u+v}\right) - \left(-e^{2u-v} \cdot e^{u+v}\right) \]Simplify the expression:\[ \text{det}(J) = 2e^{3u} - (-e^{3u}) = 2e^{3u} + e^{3u} = 3e^{3u} \]
05

Conclusion

The Jacobian determinant \(J\) of the given transformation is \( 3e^{3u} \). This expression represents how the transformation scales area elements in the \((u, v)\)-plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Partial Derivatives
Partial derivatives are crucial in multivariable calculus. They allow us to understand how a function changes along various directions. In our transformation, we have two functions, \(x = e^{2u-v}\) and \(y = e^{u+v}\), each depending on two variables, \(u\) and \(v\). To study these changes, we compute the partial derivatives of each function with respect to \(u\) and \(v\).
  • For \(x\), \(\frac{\partial x}{\partial u} = 2e^{2u-v}\) indicates how \(x\) alters with a slight change in \(u\).
  • For \(y\), \(\frac{\partial y}{\partial v} = e^{u+v}\) describes how \(y\) shifts with a small change in \(v\).
Partial derivatives provide us with the necessary building blocks to construct the Jacobian matrix, a vital tool for analyzing transformations.
Determinant
Determinants are valuable when assessing how transformations scale areas. For a 2x2 matrix like the Jacobian, the determinant can be considered as a scale factor. Our calculation \(\text{det}(J) = 3e^{3u}\) indicates the magnitude by which areas are multiplied in the \((u, v)\)-plane after transformation to the \((x, y)\)-plane.
  • A positive determinant suggests the transformation preserves orientation.
  • A value of zero would indicate areas are squished to a line or a point.
By comprehending determinants, one can assess transformation effects beyond simple numeric values and recognize their geometric significance.
Transformation Matrix
The transformation matrix, represented here by the Jacobian matrix, is formed using partial derivatives. It captures how each variable \(u\) and \(v\) impacts each transformed variable \(x\) and \(y\). Our computed Jacobian matrix:
\[ J = \begin{bmatrix} 2e^{2u-v} & -e^{2u-v} \ e^{u+v} & e^{u+v} \end{bmatrix} \]
  • The first row \([2e^{2u-v}, -e^{2u-v}]\) describes the effect of \(u\), \(v\) on \(x\).
  • The second row \([e^{u+v}, e^{u+v}]\) shows the effect of \(u\), \(v\) on \(y\).
The matrix provides a compact representation of the entire transformation process, making analysis convenient and insightful.
Multivariable Calculus
Multivariable calculus expands the principles of single-variable calculus to functions of several variables. In this context, we examine transformations involving more than one variable, such as \(u\) and \(v\). Concepts such as partial derivatives, transformation matrices, and Jacobians are fundamental here.
  • Partial derivatives allow us to see variable-specific changes.
  • The Jacobian matrix assists in evaluating and visualizing these changes effectively.
  • Understanding transformations helps describe phenomena ranging from physics to engineering.
Multivariable calculus equips us with a broader perspective, vital for real-world problem-solving involving multi-dimensional changes.

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