Question

Let \(\mathbf{F}=(f, g, h)\) be continuously differentiable in a neighborhood of \(\mathbf{P}_{0}=\left(x_{0}, y_{0}, z_{0}, u_{0}, v_{0}\right)\). \(\mathbf{F}\left(\mathbf{P}_{0}\right)=\mathbf{0},\) and $$ \left.\frac{\partial(f, g, h)}{\partial(y, z, u)}\right|_{P_{0}} \neq 0 $$ Then Theorem 6.4 .1 implies that the conditions $$ \mathbf{F}(x, y, z, u, v)=\mathbf{0}, \quad y\left(x_{0}, v_{0}\right)=u_{0} . \quad z\left(x_{0}, v_{0}\right)=z_{0} . \quad u\left(x_{0}, v_{0}\right)=u_{0} $$ determine \(y, z,\) and \(u\) as continuously differentiable functions of \((x, v)\) near \(\left(x_{0}, v_{0}\right) .\) Use Cramer's rule to express their first partial derivatives as ratios of Jacobians.

Short answer

In this exercise, we used Cramer's rule to find the partial derivatives of the functions \(y(x, v), z(x, v),\) and \(u(x, v)\) given the vector function \(\mathbf{F} = (f, g, h)\). We computed the Jacobians of the partial derivatives and expressed them as ratios of Jacobians by applying Cramer's rule. The final expressions provide the first partial derivatives of \(y, z,\) and \(u\) as continuously differentiable functions of \((x, v)\) near the point \(\left(x_{0}, v_{0}\right)\).

Step-by-step solution

1. Write down the matrix of partial derivatives of \(\mathbf{F}\) with respect to \(y, z, u\)#

First, we need to write down the matrix of the partial derivatives of \(\mathbf{F}\) with respect to \(y, z, u\). This is given by: $$ \frac{\partial(f, g, h)}{\partial(y, z, u)} = \begin{bmatrix} \frac{\partial f}{\partial y} & \frac{\partial f}{\partial z} & \frac{\partial f}{\partial u} \\ \frac{\partial g}{\partial y} & \frac{\partial g}{\partial z} & \frac{\partial g}{\partial u} \\ \frac{\partial h}{\partial y} & \frac{\partial h}{\partial z} & \frac{\partial h}{\partial u} \end{bmatrix} $$

2. Write down the partial derivatives of \(y, z,\) and \(u\) with respect to \(x\) and \(v\)#

Next, we need to find the partial derivatives of \(y, z,\) and \(u\) with respect to \(x\) and \(v\). Let's denote the functions \(y(x, v), z(x, v),\) and \(u(x, v)\) as \(y = Y(x, v), z = Z(x, v),\) and \(u = U(x, v)\), respectively. Then, we have the following partial derivatives: $$ \frac{\partial Y}{\partial x}, \frac{\partial Y}{\partial v}, \frac{\partial Z}{\partial x}, \frac{\partial Z}{\partial v}, \frac{\partial U}{\partial x}, \ \text{and} \ \frac{\partial U}{\partial v} $$

3. Calculate the Jacobians of these partial derivatives#

Now, we need to calculate the Jacobians associated with these partial derivatives. For each partial derivative, we will replace the corresponding variable in the matrix of partial derivatives with the partial derivative itself, and compute the determinant. For example, to find the Jacobian for \(\frac{\partial Y}{\partial x}\), we will replace the first column of the matrix with the vector \((\frac{\partial Y}{\partial x}, 0, 0)^T\), and then compute the determinant.

4. Apply Cramer's rule to express the partial derivatives as ratios of Jacobians#

Finally, we can apply Cramer's rule to express the partial derivatives as ratios of the Jacobians. Cramer's rule states that the solution to a linear system of equations can be expressed as the ratio of determinants. In this case, we have: $$ \frac{\partial Y}{\partial x} = \frac{\det \left[\frac{\partial(f, g, h)}{\partial(\frac{\partial Y}{\partial x}, z, u)}\right]}{\det \left[\frac{\partial(f, g, h)}{\partial(y, z, u)}\right]}, \ \frac{\partial Y}{\partial v} = \frac{\det \left[\frac{\partial(f, g, h)}{\partial(\frac{\partial Y}{\partial v}, z, u)}\right]}{\det \left[\frac{\partial(f, g, h)}{\partial(y, z, u)}\right]} $$ $$ \frac{\partial Z}{\partial x} = \frac{\det \left[\frac{\partial(f, g, h)}{\partial(y, \frac{\partial Z}{\partial x}, u)}\right]}{\det \left[\frac{\partial(f, g, h)}{\partial(y, z, u)}\right]}, \ \frac{\partial Z}{\partial v} = \frac{\det \left[\frac{\partial(f, g, h)}{\partial(y, \frac{\partial Z}{\partial v}, u)}\right]}{\det \left[\frac{\partial(f, g, h)}{\partial(y, z, u)}\right]} $$ $$ \frac{\partial U}{\partial x} = \frac{\det \left[\frac{\partial(f, g, h)}{\partial(y, z, \frac{\partial U}{\partial x})}\right]}{\det \left[\frac{\partial(f, g, h)}{\partial(y, z, u)}\right]}, \ \frac{\partial U}{\partial v} = \frac{\det \left[\frac{\partial(f, g, h)}{\partial(y, z, \frac{\partial U}{\partial v})}\right]}{\det \left[\frac{\partial(f, g, h)}{\partial(y, z, u)}\right]} $$ These expressions give us the first partial derivatives of \(y, z,\) and \(u\) as continuously differentiable functions of \((x, v)\) near \(\left(x_{0}, v_{0}\right)\) using Cramer's rule.

Key concepts

Continuously Differentiable Functions

In real analysis, a function is referred to as continuously differentiable if all of its first partial derivatives exist and are continuous over the domain of interest. This level of smoothness, often denoted as C¹, is crucial because it implies the function does not have any abrupt changes in direction or 'kinks', and its behavior can be predicted and modeled reliably near any point within its domain.

Within the context of the given exercise, the function \( \mathbf{F} \) is assumed to be continuously differentiable near the point \( \mathbf{P}_0 \), which has substantial implications. Specifically, it means that locally around \( \mathbf{P}_0 \) the functions \(y, z, \text{and} u \) can be expressed smoothly in terms of \(x \) and \(v\) - there are no sharp turns or discontinuities to worry about when calculating their derivatives or when predicting their behavior in the vicinity of \( \mathbf{P}_0 \) as implied by Theorem 6.4.1 in the text.

Partial Derivatives

The notion of partial derivatives is central to the field of multivariable calculus, which falls under the purview of real analysis. For a function of multiple variables, the partial derivative with respect to one of those variables measures how the function changes as that variable is varied while all other variables are held constant.

In relation to the exercise, partial derivatives enable understanding of how sensitive the functions \(y, z, \text{and} u\) are to changes in \(x \) or \(v\) individually. It is these sensitivities - the gradients of the function with respect to \(y, z, \text{and} u\) - that are encapsulated in the matrix of partial derivatives. They serve as the building blocks for finding the Jacobians, which are essential in applying Cramer's rule to solve for the derivatives of the desired functions in terms of \(x \) and \(v\).

Jacobians

The concept of Jacobians is integral when dealing with transformations and functions in higher dimensions. In simple terms, a Jacobian is a determinant of a matrix consisting of first-order partial derivatives. It represents the rate at which a function transforms volume in the input space to volume in the output space.

When solving the given exercise, we calculate the Jacobian by taking the determinant of the matrix of partial derivatives after substituting one of the partial derivatives of \(y, z, \text{or} u \) with respect to \(x \text{or} v\). This substitution is akin to seeing how a slight nudge in \(x\) or \(v\) causes a ripple through the system and what that implies for the output functions. The Jacobian, in this case, gives us the 'sensitivity' of the system's outputs in relation to its inputs, which is a key step in solving the problem according to Cramer's rule.

Cramer's Rule

In linear algebra, Cramer's rule is a theorem that provides an explicit expression for the solution of a system of linear equations with as many equations as unknowns, under the condition that the system's coefficient matrix is non-singular. The solution for each variable is given as the ratio of two determinants: the determinant of a matrix formed by replacing one column of the coefficient matrix with the constants from the right side of the equations, and the determinant of the original coefficient matrix.

In our exercise context, Cramer's rule is used to express the partial derivatives of \(y, z, \text{and} u\) with respect to \(x \text{and} v\) as ratios of Jacobians. This is possible because we're dealing with a system where the determinants (Jacobians) do not equal zero, as stated in the exercise. The numerator is the determinant of the modified matrix - where a column has been replaced with partial derivatives with respect to \(x \text{or} v\) - and the denominator is the determinant of the original matrix of partial derivatives. This rule provides a clear and structured method to isolate and calculate each derivative we are interested in.