Problem 31 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The Jacobi matrix is \( \begin{bmatrix} e^{x-y} & -e^{x-y} \\ e^{x+y} & e^{x+y} \end{bmatrix} \).

Step 1: Understand Jacobi Matrix

The Jacobi matrix, also known as the Jacobian matrix, of a vector-valued function is a matrix of all first-order partial derivatives of the function. For a function \(\mathbf{f}(x, y)\) detailing \(n\) functions and \(m\) variables, it is an \(n \times m\) matrix.

Step 2: Extract Functions

Extract the components of the function \(\mathbf{f}(x, y)\). Here, \(\mathbf{f}(x, y) = \begin{bmatrix} f_1(x, y) \ f_2(x, y) \end{bmatrix}\), where \(f_1(x, y) = e^{x-y}\) and \(f_2(x, y) = e^{x+y}\).

Step 3: Compute Partial Derivatives for \(f_1(x,y)\)

Find the partial derivatives of \(f_1(x, y) = e^{x-y}\). Compute \(\frac{\partial f_1}{\partial x} = e^{x-y}\) and \(\frac{\partial f_1}{\partial y} = -e^{x-y}\).

Step 4: Compute Partial Derivatives for \(f_2(x, y)\)

Find the partial derivatives of \(f_2(x, y) = e^{x+y}\). Compute \(\frac{\partial f_2}{\partial x} = e^{x+y}\) and \(\frac{\partial f_2}{\partial y} = e^{x+y}\).

Step 5: Formulate Jacobi Matrix

Construct the Jacobi matrix using the computed partial derivatives. The matrix is:\[ J_f(x, y) = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix} = \begin{bmatrix} e^{x-y} & -e^{x-y} \ e^{x+y} & e^{x+y} \end{bmatrix} \]

Key Concepts

Partial Derivatives

Partial derivatives are a key concept when dealing with multivariable functions. Simply put, they tell us how a function changes when we slightly change one variable while keeping the others constant.
In our exercise, we have a vector-valued function with two component functions. For each of these, we calculate partial derivatives with respect to each variable.
For example, with the function \( f_1(x, y) = e^{x-y} \), we compute:

• \( \frac{\partial f_1}{\partial x} = e^{x-y} \)
• \( \frac{\partial f_1}{\partial y} = -e^{x-y} \)

This gives us an understanding of how \(f_1\) changes in the direction of both \(x\) and \(y\). Similarly, for \( f_2(x,y) = e^{x+y} \):

• \( \frac{\partial f_2}{\partial x} = e^{x+y} \)
• \( \frac{\partial f_2}{\partial y} = e^{x+y} \)

Understanding and calculating these derivatives is the first essential step to building the Jacobian matrix.

Vector-Valued Functions

Vector-valued functions are fascinating because they allow us to process multiple quantities simultaneously. Instead of mapping inputs to single outputs, these functions map inputs to vectors.
In the exercise, \( \mathbf{f}(x, y) = \begin{bmatrix} e^{x-y} \ e^{x+y} \end{bmatrix} \) is a vector-valued function composed of two component functions, \( f_1(x, y) \) and \( f_2(x, y) \).
This setup is commonly encountered in fields like physics and engineering where each component might represent different physical phenomena like temperature or pressure.
The beauty of vector-valued functions is their capacity to handle complex, multidimensional scenarios by treating them as a collection of simpler, one-dimensional problems.

Jacobian Matrix

The Jacobian matrix is a significant tool in calculus and plays a crucial role in vector calculus and optimization. It essentially acts like a bridge, transforming spaces from one dimension to another.
Specifically, for a vector-valued function, the Jacobian matrix houses all the first-order partial derivatives in a tidy arrangement. This matrix provides a snapshot of how the function behaves at any given point in its domain.
From the exercise, the Jacobian matrix \( J_f(x, y) \) was derived and is:

• \[ J_f(x, y) = \begin{bmatrix} e^{x-y} & -e^{x-y} \ e^{x+y} & e^{x+y} \end{bmatrix} \]

This array of derivatives shines light on the local linear approximation of our function. In applications, a non-zero determinant of the Jacobian can indicate the absence of critical points, whereas a zero determinant does the opposite. The Jacobian, thus, is more than just a bundle of derivatives; it is a foundational piece in understanding the behavior of multivariable systems.