Question

Verify that the transformation \(u=e^{x} \cos y, v=e^{x} \sin y\) defines a one- to-one mapping of the rectangle \(R_{x y}: 0 \leq x \leq 1,0 \leq y \leq \pi / 2\) onto a region of the \(u v\)-plane and express as an iterated integral in \(u, v\) the integral $$ \iint_{R_{s x}} \frac{e^{2 x}}{1+e^{4 x} \cos ^{2} y \sin ^{2} y} d x d y . $$

Short answer

In summary, we first verified that the given transformation is one-to-one, found the Jacobian of the transformation, changed the limits of integration, and rewrote the double integral as an iterated integral in terms of the new variables \(u\) and \(v\). The transformed integral is: $$ \iint_{R_{uv}} \frac{e^{2x}}{1+u^2v^2} \, du\, dv = \int_{1}^{e}\int_{0}^{e} \frac{e^{2x}}{1+u^2v^2} \, dv\, du $$

Step-by-step solution

Verify the transformation

To check if the transformation is one-to-one, we must make sure that the first order partial derivatives are not zero simultaneously. The given transformation is: $$ u = e^x \cos y \\ v = e^x \sin y $$ Find the partial derivatives with respect to \(x\) and \(y\): $$ \frac{\partial u}{\partial x} = e^x \cos y,\quad \frac{\partial u}{\partial y} = -e^x \sin y \\ \frac{\partial v}{\partial x} = e^x \sin y,\quad \frac{\partial v}{\partial y} = e^x \cos y $$ None of these partial derivatives are zero simultaneously in the given domain, so the transformation is one-to-one.

Find the Jacobian

The Jacobian of the transformation is the determinant of the Jacobian matrix, which can be written as: $$ J = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix} = \begin{vmatrix} e^x \cos y & -e^x \sin y \\ e^x \sin y & e^x \cos y \end{vmatrix} $$ Calculate the determinant: $$ J = \left(e^x \cos y\right) \left(e^x \cos y\right) - \left(-e^x \sin y\right) \left(e^x \sin y\right) = e^{2x}(\cos^2{y} + \sin^2{y}) = e^{2x} $$

Change the limits of integration

Given the domain \(R_{xy}: 0 \leq x \leq 1, 0 \leq y \leq \frac{\pi}{2}\), we can find the new limits of integration for \(u\) and \(v\): $$ u = e^x \cos y \quad 1 \leq e^x \leq e \quad 1 \leq \cos y \leq 1 \\ v = e^x \sin y \quad 0 \leq e^x \leq e \quad 0 \leq \sin y \leq 1 $$ Thus, the new limits of integration are: $$ 1 \leq u \leq e, \quad 0 \leq v \leq e $$

Rewrite the integral using the transformation formula

Using the transformation formula and the new limits of integration, the integral can be rewritten as: $$ \iint_{R_{uv}} \frac{J}{1+u^2v^2} \, du\, dv = \int_{1}^{e}\int_{0}^{e} \frac{e^{2x}}{1+u^2v^2} \, dv\, du $$ The transformed integral is now an iterated integral in terms of \(u\) and \(v\) over the region \(R_{uv}\): $$ \iint_{R_{uv}} \frac{e^{2x}}{1+u^2v^2} \, du\, dv = \int_{1}^{e}\int_{0}^{e} \frac{e^{2x}}{1+u^2v^2} \, dv\, du $$

Key concepts

Jacobian matrix

In the study of transformations, the Jacobian matrix is a crucial concept. It provides the rate at which a transformation from one coordinate space to another occurs. Imagine a two-dimensional space face; the Jacobian matrix for a transformation is a 2x2 matrix comprising the first-order partial derivatives pertaining to each component of the transformation.

For instance, in our exercise with the transformation \(u = e^x \cos y\) and \(v = e^x \sin y\), the Jacobian matrix includes the partial derivatives:

• \(\frac{\partial u}{\partial x} = e^x \cos y\)
• \(\frac{\partial u}{\partial y} = -e^x \sin y\)
• \(\frac{\partial v}{\partial x} = e^x \sin y\)
• \(\frac{\partial v}{\partial y} = e^x \cos y\)

These partial derivatives form a matrix whose determinant, also called the Jacobian determinant, indicates the scaling factor for the area under transformation. This is particularly relevant as it helps retain volume when switching among coordinates.

change of variables

The change of variables is a concept employed to simplify the process of integration in multivariable calculus. It often involves converting the given integration bounds and variables using a new set of coordinates that might offer easier evaluation.

In our example, changing the variables from \((x, y)\) to \((u, v)\) was essential in expressing the integral in terms of \(u\) and \(v\). The transformation \(u = e^x \cos y\) and \(v = e^x \sin y\) allows such a shift. The region bounds transform accordingly from the rectangle \(R_{xy}\) to a different region in the \(uv\)-plane, rendering the integration more approachable.

This technique also takes advantage of the determinant of the Jacobian matrix which plays a part in adjusting the values inside the integral, ensuring the result remains consistent with the original volume or area.

iterated integral

An iterated integral is the process of integrating a function over multiple variables one at a time. Consider doing it in layers, where each layer symbolically peels away a variable of integration, simplifying step by step. In context, when changing variables, it becomes simpler to consider integrations in the new coordinate system.

For our problem, the transformation resulted in clearly defined new variable limits for \(u\) and \(v\), permitting the breaking down of the original double integral into iterated integrals:

• \(\int_{1}^{e}\)
• \(\int_{0}^{e}\)

This manner of integration uses a stepwise approach and focuses on repetitive integration operations enhanced by the new coordinate bounds. The integration aligns one boundary after another, efficiently tackling the computation task.

partial derivatives

Partial derivatives are central tools in multivariable calculus. They measure the rate of change of a function in response to changes in one variable while keeping others constant. Visualize them as slopes in directions orthogonal to other axes, providing insights into the curvature and behavior of surfaces.

In this exercise, the transformation \(u = e^x \cos y\) and \(v = e^x \sin y\) called for calculating the partial derivatives with respect to \(x\) and \(y\). These derivatives include:

• \(\frac{\partial u}{\partial x} = e^x \cos y\)
• \(\frac{\partial u}{\partial y} = -e^x \sin y\)
• \(\frac{\partial v}{\partial x} = e^x \sin y\)
• \(\frac{\partial v}{\partial y} = e^x \cos y\)

These values build our Jacobian matrix, ensuring that the transformation is one-to-one and valid, unlocking the path to changing variables and advancing our integral transformation efficiently.