Problem 4
Question
Evaluate \(\int_{F} K\left(x_{1}, x_{2}\right) d V_{2}(x)\), where \(F\) is bounded by the curves whose equations are given. Perform the integration by introducing variables \(u_{1}, u_{2}\) as indicated. Draw a graph of \(F\) and the corresponding region in the \(u_{1}, u_{2}\)-plane. Find the inverse of each transformation. \(K\left(x_{1}, x_{2}\right)=\left(x_{1}^{2}+x_{2}^{2}\right)^{-3} \cdot F\) is bounded by \(x_{1}^{2}+x_{2}^{2}=2 x_{1}, x_{1}^{2}+x_{2}^{2}=4 x_{1}, x_{1}^{2}+\) \(x_{2}^{2}=2 x_{2}, x_{1}^{2}+x_{2}^{2}=6 x_{2} .\) Mapping: \(x_{1}=u_{1} /\left(u_{1}^{2}+u_{2}^{2}\right), x_{2}=u_{2} /\left(u_{1}^{2}+u_{2}^{2}\right)\)
Step-by-Step Solution
Verified Answer
The correct evaluation of the integral involves a change of variables using the given mapping, calculation of the Jacobian to adjust for the change in volume elements, and integration over the transformed region, followed by finding the inverse transformation.
1Step 1: Understand the Given Functions and Mappings
Identify the given kernel function K(x1, x2) which needs to be integrated over the region F described by four curves. The mapping provided translates the Cartesian coordinates (x1, x2) into new coordinates (u1, u2) in such a way that the circular symmetry of the boundary curves of F becomes more manageable in the (u1, u2)-plane. Additionally, draw the region F and the corresponding region in the (u1, u2)-plane to understand the integration limits.
2Step 2: Draw the Region F
The boundary of region F consists of circles centered at different points. Plot the curves on a Cartesian plane to visualize the integration region F. The circles have radii incremented by the factor of sqrt(2) at each step and are centered along the lines x1=x2 and x1=2x2.
3Step 3: Perform Substitution with Given Mappings
Substitute the Cartesian coordinates (x1, x2) with the new coordinates (u1, u2) using the given mapping. This should simplify the integration process by taking advantage of the circular symmetry of the region F.
4Step 4: Find the Jacobian of the Transformation
Compute the Jacobian determinant of the transformation from (x1, x2) to (u1, u2) to account for the change of variables in the integration process.
5Step 5: Set Up the Integral in New Variables
Express the integral with the new variables u1 and u2 and the Jacobian determinant as the measure of volume. This step involves rewriting the kernel function K in terms of u1 and u2 as well.
6Step 6: Evaluate the Integral
Carry out the integral over the new region corresponding to F in the (u1, u2)-plane. Due to the mapping, this should now be simpler and possibly involve a radial and an angular part in the case of polar coordinates.
7Step 7: Find the Inverse Transformation
Finally, determine the inverse mapping from (u1, u2) back to (x1, x2). Although this may not be necessary for evaluating the integral, it can provide further insight into the relationship between the two coordinate systems.
Key Concepts
Change of Variables in IntegralsJacobian DeterminantEvaluation of IntegralsCoordinate Transformation
Change of Variables in Integrals
The method of changing variables in integrals is a powerful tool for solving complex integral problems, especially when the integrand and the region of integration have symmetries that are not easily accessible with the initial variables.
When faced with an integral over a complicated region, like the area bounded by the curves in our exercise, employing a change of variables can considerably simplify the problem. Essentially, we choose a new set of variables that transforms the region of integration into one where the limits are easier to manage. This method is crucial when integrating functions over circular, elliptical, or similarly symmetric regions. By switching from Cartesian coordinates to another coordinate system, like polar coordinates, the boundaries of the integral may become simple constants, making the integral more straightforward to evaluate.
In the exercise, the region F described by several curves is simplified by transforming from the variables \(x_1, x_2\) to \(u_1, u_2\), taking advantage of the circular symmetry of the boundary curves. The goal is to convert a tricky integration problem into a more tractable one by using this technique.
When faced with an integral over a complicated region, like the area bounded by the curves in our exercise, employing a change of variables can considerably simplify the problem. Essentially, we choose a new set of variables that transforms the region of integration into one where the limits are easier to manage. This method is crucial when integrating functions over circular, elliptical, or similarly symmetric regions. By switching from Cartesian coordinates to another coordinate system, like polar coordinates, the boundaries of the integral may become simple constants, making the integral more straightforward to evaluate.
In the exercise, the region F described by several curves is simplified by transforming from the variables \(x_1, x_2\) to \(u_1, u_2\), taking advantage of the circular symmetry of the boundary curves. The goal is to convert a tricky integration problem into a more tractable one by using this technique.
Jacobian Determinant
The Jacobian determinant is an essential concept when performing a change of variables in integrals. It represents the factor by which volume (or area, in the case of two variables) changes under a transformation of variables.
In multiple integrals, the Jacobian accounts for the distortion that occurs when switching from one set of variables to another. If you think about stretching or compressing a rubber sheet, the Jacobian tells you how much a small patch of area on the sheet changes as it is distorted. In formal terms, it is the absolute value of the determinant of the matrix of first-order partial derivatives of the transformed variables with respect to the original variables.
For the transformation given in the exercise, we would compute the determinant of the matrix formed by the partial derivatives of \(x_1\) and \(x_2\) with respect to \(u_1\) and \(u_2\). This resulting determinant, the Jacobian, is then used as a multiplicative factor in the integral to reflect the stretching or compressing of the area elements during the transformation.
In multiple integrals, the Jacobian accounts for the distortion that occurs when switching from one set of variables to another. If you think about stretching or compressing a rubber sheet, the Jacobian tells you how much a small patch of area on the sheet changes as it is distorted. In formal terms, it is the absolute value of the determinant of the matrix of first-order partial derivatives of the transformed variables with respect to the original variables.
For the transformation given in the exercise, we would compute the determinant of the matrix formed by the partial derivatives of \(x_1\) and \(x_2\) with respect to \(u_1\) and \(u_2\). This resulting determinant, the Jacobian, is then used as a multiplicative factor in the integral to reflect the stretching or compressing of the area elements during the transformation.
Evaluation of Integrals
The evaluation of integrals is often the ultimate step in the process of solving an area or volume problem in calculus. It involves calculating the definite integral of a function over a particular region, which requires setting appropriate limits of integration and finding the antiderivative of the integrand.
The steps we follow typically include identifying the shape of the region, choosing a method of integration (such as substitution or parts), determining the limits of integration, and calculating the integral. In challenging cases where direct integration is difficult, changing variables as we did in the exercise may lead to an easier problem. After employing the Jacobian to account for the change of variables, the integral must be evaluated with respect to the new variables, considering the transformed limits and integrand. The process might also involve several steps of integration, depending on whether the integral is single, double, or triple.
The steps we follow typically include identifying the shape of the region, choosing a method of integration (such as substitution or parts), determining the limits of integration, and calculating the integral. In challenging cases where direct integration is difficult, changing variables as we did in the exercise may lead to an easier problem. After employing the Jacobian to account for the change of variables, the integral must be evaluated with respect to the new variables, considering the transformed limits and integrand. The process might also involve several steps of integration, depending on whether the integral is single, double, or triple.
Coordinate Transformation
Coordinate transformation is the process of moving from one coordinate system to another to facilitate the evaluation of integrals, especially in multivariable calculus. Coordinates are merely a way to describe points in a space, and different coordinate systems can be better suited to different problems.
In our exercise, we perform a transformation from the original Cartesian coordinates \(x_1\) and \(x_2\) to new coordinates \(u_1\) and \(u_2\) to exploit the symmetry of the problem. This kind of transformation is particularly useful for regions with circular or spherical symmetry. Other common transformations include polar, cylindrical, and spherical coordinates, each helpful in different situations. Performing such a transformation typically involves not just changing the variables but also computing the Jacobian to compensate for the change in scale.
The transformation allows us to rewrite the integral in a more manageable form, where the geometry of the problem becomes simpler, and often, the integration process becomes more straightforward.
In our exercise, we perform a transformation from the original Cartesian coordinates \(x_1\) and \(x_2\) to new coordinates \(u_1\) and \(u_2\) to exploit the symmetry of the problem. This kind of transformation is particularly useful for regions with circular or spherical symmetry. Other common transformations include polar, cylindrical, and spherical coordinates, each helpful in different situations. Performing such a transformation typically involves not just changing the variables but also computing the Jacobian to compensate for the change in scale.
The transformation allows us to rewrite the integral in a more manageable form, where the geometry of the problem becomes simpler, and often, the integration process becomes more straightforward.
Other exercises in this chapter
Problem 3
Show that the relation \(F(x, y)=0\) yields \(y\) as a function of \(x\) in an interval \(I\) about \(x_{0}\) where \(F\left(x_{0}, y_{0}\right)=0\). Denote the
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View solution Problem 4
Show that the relation \(F(x, y)=0\) yields \(y\) as a function of \(x\) in an interval \(I\) about \(x_{0}\) where \(F\left(x_{0}, y_{0}\right)=0\). Denote the
View solution Problem 5
Find the solution by the Lagrange multiplier rule. Find the minimum value of \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}\) subject to the condition \(2 x_{1}+\) \
View solution