Question

In Exercises \(27-30,\) find the Jacobian \(\partial(x, y, z) / \partial(u, v, w)\) for the indicated change of variables. If \(x=f(u, v, w), y=g(u, v, w)\) and \(z=h(u, v, w),\) then the Jacobian of \(x, y,\) and \(z\) with respect to \(u, v,\) and \(w\) is \(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right| .\) \(x=4 u-v, y=4 v-w, z=u+w\)

Short answer

The Jacobian determinant for the given transformation is 16.

Step-by-step solution

Identifying the Transformation Equations

Initially, identify the transformation equations given by the problem. In this case, they are defined as: \(x=4u-v\), \(y=4v-w\), \(z=u+w\).

Calculate the Partial Derivatives

To form the Jacobian matrix, it's necessary to calculate the partial derivatives of \(x\), \(y\), and \(z\) with respect to \(u\), \(v\), and \(w\). You will get: \(\frac{\partial x}{\partial u} = 4\), \(\frac{\partial x}{\partial v} = -1\), \(\frac{\partial x}{\partial w} = 0\), \(\frac{\partial y}{\partial u} = 0\), \(\frac{\partial y}{\partial v} = 4\), \(\frac{\partial y}{\partial w} = -1\), \(\frac{\partial z}{\partial u} = 1\), \(\frac{\partial z}{\partial v} = 0\), and \(\frac{\partial z}{\partial w} = 1\).

Create the Jacobian Matrix and Evaluate the Determinant

Now, replace these values into the general formula for the Jacobian determinant. The Jacobian matrix will look like this: \[\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}4 & -1 & 0 \ 0 & 4 & -1 \ 1 & 0 & 1\end{array}\right|.\]. Then, calculate the determinant of this 3x3 matrix. The result is \(4*4*1 - 1*-1*0 = 16\).

Key concepts

Partial Derivatives

Understanding partial derivatives is essential when dealing with functions that depend on multiple variables, which is exactly what we encounter in multivariable calculus. When you calculate a partial derivative, you focus on the rate of change of a function with respect to one of its variables, while treating all other variables as constants.

Take a function such as \( f(x, y, z)=4u-v \), which depends on the variables \( u \), \( v \), and \( w \). The partial derivative \( \frac{\partial f}{\partial u} \) represents the sensitivity of the function to changes in \( u \) while holding \( v \) and \( w \) fixed. In our exercise, computing the partial derivatives of \( x \) with respect to \( u \) gives us \( 4 \) because \( x = 4u - v \) changes by four units for each increase in one unit of \( u \) and \( v \) doesn’t affect this rate as it is treated as a constant in this calculation. Similarly, all other partial derivatives are calculated while keeping the other variables constant.

Transformation Equations

Transformation equations describe how a set of variables can be mapped to another set through specific functions. These equations play a pivotal role in changing the perspective with which we perceive certain mathematical problems, often simplifying complex calculations or providing better insight into the problem's geometry.

In the context of multivariable calculus, these transformations can radically simplify the integration process by changing to a coordinate system where the limits of integration become manageable or the equation of the curve or surface becomes simpler. The specific transformation given in the exercise, \( x = 4u-v \) and so on, represents a new way to express the original variables \( x, y, z \) in terms of the new variables \( u, v, w \). By identifying the transformation equations, we lay the groundwork for calculating the Jacobian determinant, which tells us how much volume is stretched or compressed when moving from one set of variables to another.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus (such as derivatives and integrals) to functions that have more than one variable. It allows us to explore spaces that are more than one-dimensional, and it's applicable in fields like physics, engineering, economics, and beyond. The Jacobian determinant, which we are calculating here, is a function of several variables and originates from the discipline of multivariable calculus.

It encapsulates information about the volume distortion caused by a transformation from one coordinate system to another. Formally, the Jacobian is the determinant of a matrix comprised of first-order partial derivatives. The problems and solutions involving the Jacobian matrix and determinant not only require calculation skills but also a deep understanding of the relationships between different coordinate systems and how they affect the properties of space, especially concerning transformations like rotations, translations, and scalings.

In the given exercise, the Jacobian matrix is built from the partial derivatives of the transformation functions. Taking the determinant of this matrix gives us a scalar value that signifies how a small volume around a point in the original space is distorted when transformed into the new space characterized by variables \( u, v, \text{and } w\).