Problem 33 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The Jacobian matrix is \[ \begin{bmatrix} -\sin(x-y) & \sin(x-y) \\ -\sin(x+y) & -\sin(x+y) \end{bmatrix} \].

Step 1: Understanding the function

The function given is a vector-valued function with two components: \[ f_1(x, y) = \cos(x-y) \] \[ f_2(x, y) = \cos(x+y) \] Our task is to find the Jacobian matrix of these functions with respect to the variables \(x\) and \(y\).

Step 2: Partial derivatives of the first component

First, we find the partial derivatives of the first component \( f_1(x, y) = \cos(x-y) \):- Partial derivative with respect to \(x\): \[ \frac{\partial f_1}{\partial x} = -\sin(x-y) \]- Partial derivative with respect to \(y\): \[ \frac{\partial f_1}{\partial y} = \sin(x-y) \]

Step 3: Partial derivatives of the second component

Next, we find the partial derivatives of the second component \( f_2(x, y) = \cos(x+y) \):- Partial derivative with respect to \(x\): \[ \frac{\partial f_2}{\partial x} = -\sin(x+y) \]- Partial derivative with respect to \(y\): \[ \frac{\partial f_2}{\partial y} = -\sin(x+y) \]

Step 4: Constructing the Jacobian matrix

The Jacobian matrix \( J \) is constructed by placing the partial derivatives into the matrix structure:\[J = \begin{bmatrix}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{bmatrix}= \begin{bmatrix}-\sin(x-y) & \sin(x-y) \-\sin(x+y) & -\sin(x+y)\end{bmatrix}\]

Key Concepts

Partial Derivatives

Partial derivatives are an extension of the concept of derivatives to functions of several variables. Imagine you have a function that depends on more than one variable, like our function \( f(x, y) \). We want to see how the function changes when we tweak just one of the variables while keeping the others constant. This is exactly what a partial derivative does.

For instance, in our exercise, the first component \( f_1(x, y) = \cos(x-y) \) requires us to calculate two partial derivatives:

• With respect to \( x \): We treat \( y \) as a constant and differentiate, resulting in \( \frac{\partial f_1}{\partial x} = -\sin(x-y) \).
• With respect to \( y \): We treat \( x \) as a constant and differentiate, giving us \( \frac{\partial f_1}{\partial y} = \sin(x-y) \).

Similarly, for \( f_2(x, y) = \cos(x+y) \), we find:

• \( \frac{\partial f_2}{\partial x} = -\sin(x+y) \).
• \( \frac{\partial f_2}{\partial y} = -\sin(x+y) \).

Understanding partial derivatives helps us understand the rate and direction of change of multi-variable functions, which is crucial in constructing the Jacobian matrix.

Vector-Valued Function

A vector-valued function, unlike a regular function that gives us a single scalar output, offers a vector as its output. Think of each input of variables mapping to multiple components, each a function in its own right.

The function \( \mathbf{f}(x, y) \) in our exercise is vector-valued because it consists of two components: \( f_1(x, y) \) and \( f_2(x, y) \). Each component is responsible for one part of the output vector.

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

Such functions are highly common in physics and engineering, where you often come across scenarios requiring multiple simultaneous outputs, like force or electric field vectors. Understanding how to deal with vector-valued functions, including differentiation and integration, is important as it underpins key calculations and simulations in many scientific fields.

Jacobi Matrix

The Jacobi matrix, also known as the Jacobian matrix in many contexts, is essentially a tool for understanding how a vector-valued function changes near a point. It encapsulates all of the partial derivatives for each function component with respect to each variable.

In our exercise, this is organized into a matrix, \( J \), as follows:

• The first row contains the partial derivatives of the first function component, \( f_1 \), with respect to the variables \( x \) and \( y \).
• The second row contains the partial derivatives of the second component, \( f_2 \), with respect to the same variables.

Putting it all together for \( f(x, y) \):\[J = \begin{bmatrix}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{bmatrix}= \begin{bmatrix}-\sin(x-y) & \sin(x-y) \-\sin(x+y) & -\sin(x+y)\end{bmatrix}\]The Jacobian matrix plays an essential role in various applications, including optimization, where it helps understand sensitivity, and in differential equations, where it assists in expressing complex systems.