Problem 29 - Calculus for Biology and Medicine

Question

Find the Jacobi matrix for each given function. \(\mathbf{f}(x, y)=\left[\begin{array}{c}x+y \\ x^{2}-y^{2}\end{array}\right]\)

Step-by-Step Solution

Answer

The Jacobian matrix is \( \begin{bmatrix} 1 & 1 \\ 2x & -2y \end{bmatrix} \).

Step 1: Identify Partial Derivatives

To find the Jacobian matrix of a vector function \( \mathbf{f}(x, y) \), we need to compute partial derivatives. The function is \( \mathbf{f}(x, y) = \begin{bmatrix} x+y \ x^2-y^2 \end{bmatrix} \). We identify each component of \( \mathbf{f} \): \( f_1(x, y) = x+y \) and \( f_2(x, y) = x^2-y^2 \).

Step 2: Calculate Partial Derivatives for f1

For \( f_1(x, y) = x+y \), compute the partial derivatives with respect to \( x \) and \( y \): \( \frac{\partial f_1}{\partial x} = 1 \) and \( \frac{\partial f_1}{\partial y} = 1 \).

Step 3: Calculate Partial Derivatives for f2

For \( f_2(x, y) = x^2-y^2 \), calculate the partial derivatives: \( \frac{\partial f_2}{\partial x} = 2x \) and \( \frac{\partial f_2}{\partial y} = -2y \).

Step 4: Form the Jacobian Matrix

The Jacobian matrix \( J \) is composed of the partial derivatives we calculated: \[ J = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix} = \begin{bmatrix} 1 & 1 \ 2x & -2y \end{bmatrix} \]

Key Concepts

Understanding Partial Derivatives

The foundation of the Jacobian matrix lies in the concept of partial derivatives. Partial derivatives are used to understand how a multivariable function changes with respect to one of its variables while keeping the other variables constant. This is particularly useful when dealing with functions that have more than one input, such as our vector function \( \mathbf{f}(x, y) = \begin{bmatrix} x+y \ x^2-y^2 \end{bmatrix} \). To find the partial derivative of a function with respect to a specific variable, we treat all other variables as constants and differentiate with respect to the chosen variable.

This approach lets us measure the rate of change in one direction while ignoring all others. For example, in \( f_1(x, y) = x+y \), the partial derivative with respect to \( x \), denoted by \( \frac{\partial f_1}{\partial x} \), is calculated by differentiating \( x+y \) concerning \( x \), resulting in \( 1 \). Similarly, \( \frac{\partial f_1}{\partial y} \) is also \( 1 \) because when we differentiate with respect to \( y \), \( x \) is treated as a constant. Partial derivatives thus provide a snapshot of how a function behaves along each axis individually.

Introduction to Vector Functions

Vector functions play a vital role in multivariable calculus. A vector function takes one or more variables as input and produces a vector as output. In our specific case, the vector function \( \mathbf{f}(x, y) \) maps variables \( x \) and \( y \) to a two-dimensional vector. The vector function can be broken down into separate components that represent different aspects of the complete output.

Each component of a vector function is typically a scalar function on its own, such as:

• \( f_1(x, y) = x+y \)
• \( f_2(x, y) = x^2-y^2 \)

By examining these components, we can analyze how each one contributes to the vector's final form. This decomposition is essential for many calculations, especially when forming and understanding matrices like the Jacobian. It assists in capturing not only the magnitude but also the direction of changes caused by variations in the input variables.

Formation of the Jacobian Matrix

The Jacobi or Jacobian matrix is a crucial entity in calculus for vector functions. It is essentially a matrix that organizes all the first-order partial derivatives of a vector function. To construct the Jacobian matrix for a vector function like \( \mathbf{f}(x, y) = \begin{bmatrix} x+y \ x^2-y^2 \end{bmatrix} \), we must first compute all the necessary partial derivatives.

For each component function, we compute the partial derivative with respect to each input variable. The partial derivatives we've already computed contribute to the formation of this matrix. For example: - The partial derivatives of \( f_1(x, y) \) are: \( \frac{\partial f_1}{\partial x} = 1 \) and \( \frac{\partial f_1}{\partial y} = 1 \). - The partial derivatives of \( f_2(x, y) \) are: \( \frac{\partial f_2}{\partial x} = 2x \) and \( \frac{\partial f_2}{\partial y} = -2y \).
These derivatives are then structured into a matrix: \[J = \begin{bmatrix} 1 & 1 \ 2x & -2y \end{bmatrix}\] The Jacobian matrix provides insights into how small changes in the input variables will affect the output vector, serving as a bridge between an input space and its corresponding output.