Problem 34 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

Jacobi matrix is \[ J = \begin{bmatrix} \frac{1}{x+y} & \frac{1}{x+y} \\ e^{x+y} & e^{x+y} \end{bmatrix} \]

Step 1: Understand the Function

We are given a vector function, \( \mathbf{f}(x, y) = \left[ \begin{array}{c} \ln(x+y) \ e^{x+y} \end{array} \right] \). The goal is to find the Jacobi matrix of this function, which consists of partial derivatives of each component of the function with respect to each variable.

Step 2: Differentiate the First Component

The first component of the function is \( f_1(x, y) = \ln(x+y) \). The partial derivative of \( f_1 \) with respect to \( x \) is \( \frac{\partial}{\partial x} \ln(x+y) = \frac{1}{x+y} \). Similarly, the partial derivative with respect to \( y \) is \( \frac{\partial}{\partial y} \ln(x+y) = \frac{1}{x+y} \).

Step 3: Differentiate the Second Component

The second component of the function is \( f_2(x, y) = e^{x+y} \). The partial derivative of \( f_2 \) with respect to \( x \) is \( \frac{\partial}{\partial x} e^{x+y} = e^{x+y} \). Similarly, the partial derivative with respect to \( y \) is \( \frac{\partial}{\partial y} e^{x+y} = e^{x+y} \).

Step 4: Construct the Jacobi Matrix

The Jacobi matrix \( J \) is constructed using the partial derivatives. For the given function, it's a 2x2 matrix given by: \[J = \begin{bmatrix}\frac{1}{x+y} & \frac{1}{x+y} \e^{x+y} & e^{x+y}\end{bmatrix}\]

Key Concepts

Understanding Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They are essentially the derivatives of functions with more than one variable. When we perform a partial derivative, we are focusing on how a function changes in response to changes in just one variable, while keeping the others constant.
For example, if we have a function \( f(x, y) \), and we take the partial derivative with respect to \( x \), we observe how the function varies as \( x \) changes and \( y \) remains fixed. The symbolism for this is \( \frac{\partial f}{\partial x} \).
In the exercise, the partial derivative of the first function component \( \ln(x+y) \) with respect to \( x \) and \( y \) was found to be \( \frac{1}{x+y} \). Similarly, the second function component \( e^{x+y} \) shows the partial derivatives \( e^{x+y} \) for both \( x \) and \( y \).

• This demonstrates how each variable affects the outcome of the function individually.
• Partial derivatives help in understanding the gradient or slope in various directions of multivariable functions.

Understanding this concept is crucial for constructing higher-dimensional derivations like the Jacobi matrix.

Exploring Vector Functions

A vector function is a function that produces a vector as its output. In multivariable calculus, vector functions are often used to describe physical and geometric phenomena. Each component of a vector function is a scalar function of the input variables.
In our exercise, the vector function \( \mathbf{f}(x, y) \) has two components, \( \ln(x+y) \) and \( e^{x+y} \). This means for each pair of \( (x, y) \), the functions can be evaluated to produce a vector in 2-dimensional space.

• Vector functions can represent concepts like velocity, force, and electric field, where multiple dimensions and factors are considered.
• They allow multiple outputs to be calculated simultaneously using a single input.

Analyzing vector functions involves understanding each component function separately, which we then use to build comprehensive solutions, such as the Jacobi matrix.

Introducing Matrix Calculus and the Jacobi Matrix

Matrix calculus extends the concepts of calculus, such as derivatives, to matrices, which are collections of numbers arranged into rows and columns. The Jacobi matrix is an essential part of matrix calculus and is used primarily in multivariable calculus.
In simple terms, a Jacobi matrix is a matrix of all first-order partial derivatives of a vector function. For a function \( \mathbf{f}(x, y) \) as given in the exercise, the Jacobi matrix compiles all the partial derivatives to express how changes in each variable affect each component of the function.
The matrix appears structured as follows for our specific problem:

• A 2x2 matrix where each entry is a partial derivative \[\begin{bmatrix}\frac{1}{x+y} & \frac{1}{x+y} \ e^{x+y} & e^{x+y}\end{bmatrix}\].
• Each row represents how a single component of the vector function changes with respect to each variable.

Understanding and constructing the Jacobi matrix is crucial for performing more complex analyses, such as optimization problems and studying the stability of systems. This matrix provides a comprehensive snapshot of the derivative behaviour across different dimensions, which is instrumental in advanced engineering, physics, and mathematics.