Chapter 13: Problem 13
Find the transformation from the \(u v\) -plane to the xy-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x^{2}+y^{2}, v=x $$
Short Answer
Expert verified
The Jacobian determinant is \(-\frac{1}{2\sqrt{u-v^2}}\).
Step by step solution
01
Express x and y in terms of u and v
We have the equations \(u = x^2 + y^2\) and \(v = x\). We can express \(x\) directly as \(x = v\). Substitute \(v\) for \(x\) in the first equation to find \(y^2\):\[u = v^2 + y^2\]. Hence, \(y^2 = u - v^2\), leading to \(y = \sqrt{u - v^2}\).
02
Derive the Jacobian Matrix
The Jacobian Matrix \(J\) of the transformation from the \(uv\)-plane to the \(xy\)-plane is the matrix of partial derivatives: \[J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}.\] Since \(x = v\), we have:\[\frac{\partial x}{\partial u} = 0, \quad \frac{\partial x}{\partial v} = 1.\] For \(y = \sqrt{u - v^2}\), the derivatives are:\[\frac{\partial y}{\partial u} = \frac{1}{2\sqrt{u - v^2}}, \quad \frac{\partial y}{\partial v} = -\frac{v}{\sqrt{u - v^2}}.\]
03
Calculate the Jacobian Determinant
The determinant of the Jacobian matrix \(J\) is calculated as follows:\[|J| = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \begin{vmatrix} 0 & 1 \ \frac{1}{2\sqrt{u - v^2}} & -\frac{v}{\sqrt{u - v^2}} \end{vmatrix} = \left(0 \cdot -\frac{v}{\sqrt{u - v^2}}\right) - \left(1 \cdot \frac{1}{2\sqrt{u - v^2}}\right) = -\frac{1}{2\sqrt{u-v^2}}.\]
04
Conclusion on Jacobian
The negative sign in the Jacobian determinant indicates that the transformation involves a reflection, possibly through varying orientation. The Jacobian \(|J|\) reveals the local scaling factor of area transformation near each point in the \(uv\)-plane to the \(xy\)-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 an essential concept in calculus, particularly when dealing with functions of multiple variables. When you have a function such as \( f(x, y) \), the partial derivative of \( f \) with respect to \( x \), denoted as \( \frac{\partial f}{\partial x} \), measures how \( f \) changes as \( x \) changes, while keeping \( y \) constant. Similarly, \( \frac{\partial f}{\partial y} \) measures how \( f \) changes with respect to \( y \) while keeping \( x \) constant.
In the context of transformations, partial derivatives help to construct the Jacobian matrix. This matrix is formed by taking partial derivatives of the coordinate transformation functions with respect to each of the variables involved. These derivatives will then illustrate how small changes in one set of coordinates affect another.
Partial derivatives provide a detailed look at how each component of a vector or transformation changes independently, aiding in the analysis of complex multivariable systems.
In the context of transformations, partial derivatives help to construct the Jacobian matrix. This matrix is formed by taking partial derivatives of the coordinate transformation functions with respect to each of the variables involved. These derivatives will then illustrate how small changes in one set of coordinates affect another.
Partial derivatives provide a detailed look at how each component of a vector or transformation changes independently, aiding in the analysis of complex multivariable systems.
Coordinate Transformation
A coordinate transformation is a mathematical operation that helps to convert coordinates from one system into another. In our example, we are trying to transform coordinates from the \( uv \)-plane to the \( xy \)-plane. The specific transformation equations given are \( u = x^2 + y^2 \) and \( v = x \).
Coordinate transformations are widely used in various fields, such as physics and engineering, to simplify complex problems or to convert them into a more useful or applicable coordinate system. They help in understanding how a geometric shape or space looks like when represented in a different plane.
During transformations, it's essential to ensure that every point in one plane corresponds correctly to a point in the other plane. This involves finding explicit expressions for the new coordinates in terms of the old ones, which was achieved in the solution by expressing \( x \) and \( y \) in terms of \( u \) and \( v \).
Coordinate transformations are widely used in various fields, such as physics and engineering, to simplify complex problems or to convert them into a more useful or applicable coordinate system. They help in understanding how a geometric shape or space looks like when represented in a different plane.
During transformations, it's essential to ensure that every point in one plane corresponds correctly to a point in the other plane. This involves finding explicit expressions for the new coordinates in terms of the old ones, which was achieved in the solution by expressing \( x \) and \( y \) in terms of \( u \) and \( v \).
Jacobian Matrix
The Jacobian matrix is a critical tool in calculus for dealing with multivariable transformations. It consists of all the first-order partial derivatives of a vector-valued function. For example, if you have a transformation from the \( uv \)-plane to the \( xy \)-plane, the Jacobian matrix \( J \) is structured as:
\[J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}. \]
This matrix allows us to understand how infinitesimal changes in \( u \) and \( v \) affect \( x \) and \( y \).
In practical applications, the Jacobian matrix is used to linearize a transformation around points of interest, providing insights into the local behavior of the transformation. This is especially useful in optimizing systems or integrating over transformed spaces.
\[J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}. \]
This matrix allows us to understand how infinitesimal changes in \( u \) and \( v \) affect \( x \) and \( y \).
In practical applications, the Jacobian matrix is used to linearize a transformation around points of interest, providing insights into the local behavior of the transformation. This is especially useful in optimizing systems or integrating over transformed spaces.
Determinant
The determinant of the Jacobian matrix, often simply called "the Jacobian," plays a vital role in analyzing transformations between coordinate systems. For a 2x2 Jacobian matrix \( J \), its determinant can be computed as:
\[ |J| = \frac{\partial x}{\partial u} \cdot \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \cdot \frac{\partial y}{\partial u}. \]
In our exercise, the determinant is \(-\frac{1}{2\sqrt{u-v^2}}\). This value is important because it indicates how the transformation scales areas as it moves from one coordinate plane to another.
The sign of the determinant also tells us about the orientation change. A negative determinant implies that the transformation includes a reflection, meaning that the original and transformed planes may have reversed orientations.
\[ |J| = \frac{\partial x}{\partial u} \cdot \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \cdot \frac{\partial y}{\partial u}. \]
In our exercise, the determinant is \(-\frac{1}{2\sqrt{u-v^2}}\). This value is important because it indicates how the transformation scales areas as it moves from one coordinate plane to another.
The sign of the determinant also tells us about the orientation change. A negative determinant implies that the transformation includes a reflection, meaning that the original and transformed planes may have reversed orientations.
Area Scaling
Area scaling is a concept closely tied to the determinant of the Jacobian. It tells us by what factor areas are stretched or compressed when transformed from one coordinate system to another. Specifically, the absolute value of the determinant \(|J|\) represents this scaling factor.
Consider a small area in the \( uv \)-plane. Under transformation, this area morphs to a new area in the \( xy \)-plane. The magnitude of the Jacobian determinant at a given point highlights how the size of the small area changes. A determinant greater than one indicates an expansion, while a determinant less than one indicates a contraction.
This concept is particularly crucial in integration and analyzing geometric transformations, as it allows us to correct for changes in size when migrating between coordinate systems. Understanding area scaling helps in applications such as computing volumes under surfaces or evaluating how physical phenomena vary under different spatial conditions.
Consider a small area in the \( uv \)-plane. Under transformation, this area morphs to a new area in the \( xy \)-plane. The magnitude of the Jacobian determinant at a given point highlights how the size of the small area changes. A determinant greater than one indicates an expansion, while a determinant less than one indicates a contraction.
This concept is particularly crucial in integration and analyzing geometric transformations, as it allows us to correct for changes in size when migrating between coordinate systems. Understanding area scaling helps in applications such as computing volumes under surfaces or evaluating how physical phenomena vary under different spatial conditions.
