Question

Suppose that \(\mathbf{A}=\left[a_{i j}\right]\) is a nonsingular \(n \times n\) matrix and \(T\) is the region in \(\mathbb{R}^{n}\) defined by $$ \alpha_{1} \leq a_{i 1} x_{1}+a_{i 2} x_{2}+\cdots+a_{i n} x_{n} \leq \beta_{i}, \quad 1 \leq i \leq n. $$ (a) Find \(V(T)\). (b) Show that if \(c_{1}, c_{2}, \ldots, c_{n}\) are constants, then $$ \int_{T}\left(\sum_{j=1}^{n} c_{j} x_{j}\right) d \mathbf{X}=\frac{V(T)}{2} \sum_{i=1}^{n} d_{i}\left(\alpha_{i}+\beta_{i}\right), $$ where $$ \left[\begin{array}{c} d_{1} \\ d_{2} \\ \vdots \\ d_{n} \end{array}\right]=\left(\mathbf{A}^{t}\right)^{-1}\left[\begin{array}{c} c_{1} \\ c_{2} \\ \vdots \\ c_{n} \end{array}\right]. $$

Short answer

Question: Compute the integral over region $T$ of the sum of $n$ constants times the coordinates, where $T$ is a region in $\mathbb{R}^n$ defined by a set of inequalities and has a nonsingular $n \times n$ matrix $\mathbf{A}$. Answer: The integral can be calculated as $$ \int_{T}\left(\sum_{j=1}^{n} c_{j} x_{j}\right) d \mathbf{X} = \frac{V(T)}{2} \sum_{i=1}^{n} d_{i}\left(\alpha_{i}+\beta_{i}\right), $$ where $V(T) = |\mathbf{A}|^{-1} \prod_{i=1}^{n} (\beta_{i} - \alpha_{i})$ and $\mathbf{d} = (\mathbf{A}^t)^{-1}\mathbf{c}$.

Step-by-step solution

Find the volume of region \(T\)#

First, we will find the volume \(V(T)\) of the region \(T\), which can be done by applying a change of variables. Let \(\mathbf{Y} = \mathbf{A}\mathbf{X}\). Then the Jacobian matrix is \(\mathbf{A}\), and its determinant is \(J(\mathbf{X}) = |\mathbf{A}|\). Therefore, the inequalities defining region \(T\) can be rewritten as $$ \alpha_{1} \leq y_{1} \leq \beta_{1}, \dots, \alpha_{n} \leq y_{n} \leq \beta_{n}. $$ This new region is a parallelepiped, and its volume is given by $$ V(T) = \int_{T'} d\mathbf{Y} = \prod_{i=1}^{n} (\beta_{i} - \alpha_{i}) = \left|\mathbf{A}\right|^{-1} \prod_{i=1}^{n} (\beta_{i} - \alpha_{i}). $$

Calculate the integral#

Now we can calculate the integral over region \(T\). Using the change of variables we derived in step 1, we get: $$ \begin{aligned} \int_{T}\left(\sum_{j=1}^{n} c_{j} x_{j}\right) d \mathbf{X}&=\int_{T}\left(\mathbf{c}^T\mathbf{X}\right) d \mathbf{X}\\ &=\int_{T'}\left(\mathbf{c}^T\mathbf{A}^{-1}\mathbf{Y}\right) |\mathbf{A}| d \mathbf{Y}\\ &=\int_{T'}\left(\mathbf{d}^T\mathbf{Y}\right) d \mathbf{Y}, \end{aligned} $$ where \(\mathbf{d} = (\mathbf{A}^t)^{-1}\mathbf{c}\). Now, the integral can be rewritten as a product of one-dimensional integrals: $$ \int_{T'}\left(\mathbf{d}^T\mathbf{Y}\right) d \mathbf{Y} = \prod_{i=1}^{n} \int_{\alpha_i}^{\beta_i} d_i y_i dy_i. $$ Evaluating the integrals, we get $$ \int_{T'}\left(\mathbf{d}^T\mathbf{Y}\right) d \mathbf{Y} = \prod_{i=1}^{n} \frac{d_i}{2}(\beta_i^2 - \alpha_i^2). $$ Then, substituting the expression for \(V(T)\), $$ \int_{T}\left(\sum_{j=1}^{n} c_{j} x_{j}\right) d \mathbf{X}=\frac{V(T)}{2} \sum_{i=1}^{n} d_{i}\left(\alpha_{i}+\beta_{i}\right). $$

Key concepts

Matrix Transformations

Matrix transformations involve changing variables or coordinates in multivariable calculus. When you have a system described by multiple variables, transformations allow you to express these variables using linear combinations of new variables. This is represented using matrices.

• Matrix Definition: A matrix is a rectangular array of numbers arranged in rows and columns. For example, an \( n \times n \) matrix has \( n \) rows and \( n \) columns.
• Purpose: They help simplify calculations, especially when dealing with linear transformations from one space to another. This is particularly useful when analyzing shapes like regions in space.
• Transformation Function: In our context, a transformation with a matrix \( \mathbf{A} \) is applied to a vector \( \mathbf{X} \) by multiplying them: \( \mathbf{Y} = \mathbf{A} \mathbf{X} \).

Matrix transformations are essential for restructuring regions, such as changing a region \( T \) defined in a matrix system into a simpler form for easier volume computation.

Jacobian Matrix

The Jacobian matrix is a critical component when dealing with multivariable calculus transformations. It represents the gradient of a vector-valued function and helps in changing variables during integration.

• Definition: The Jacobian is a matrix of all first-order partial derivatives of a vector function. For a function \( f: \mathbb{R}^n \to \mathbb{R}^m \), the Jacobian \( J \) would be an \( m \times n \) matrix.
• Role in Transformations: When transforming regions, the determinant of the Jacobian matrix helps compute how much a region scales during a transformation. If \( \mathbf{Y} = \mathbf{A}\mathbf{X} \), then the Jacobian of this transformation is simply \( \mathbf{A} \).
• Usefulness: Its determinant gives us the factor by which areas (or volumes) are scaled when variables are transformed. For example, in our exercise, \( \det(\mathbf{A}) = |\mathbf{A}| \) represents this scaling factor.

This matrix is particularly useful for changing variables in integrals, simplifying the process of calculating volumes and other integrals over transformed domains.

Volume of a Region

Calculating the volume of a region is a vital application of multivariable integration. When a region in space is transformed via a matrix, its volume can be determined by accounting for the transformation's scaling effect.

• Region Definition: A region \( T \) in \( \mathbb{R}^n \) is defined by inequalities involving linear combinations of variables. These can represent physical space constraints.
• Parallelepiped Volume: After a transformation, a complex region may become a simpler shape, like a parallelepiped. The volume of such a shape in a new space can be easily determined.
• Integration Techniques: The volume of the new region \( T' \) can be found using integration. For axis-aligned regions defined by \( \alpha_i \leq y_i \leq \beta_i \), volume is calculated as \( V(T') = \prod_{i=1}^{n} (\beta_i - \alpha_i) \).
• Adjustment for Transformation: To find the original region's volume, adjust for the transformation matrix's determinant by dividing by \( |\mathbf{A}| \). Thus, the volume \( V(T) = \frac{1}{|\mathbf{A}|} \prod_{i=1}^{n} (\beta_i - \alpha_i) \).

Understanding how volume adjusts during transformations is essential for solving integrals over complex regions and ensures that calculations are precise and reflective of the geometric transformations undergone.