Problem 30 - Calculus for Biology and Medicine

Question

Find the Jacobi matrix for each given function. \(\mathbf{f}(x, y)=\left[\begin{array}{c}2 x-3 y \\ 4 x^{2}\end{array}\right]\)

Step-by-Step Solution

Answer

The Jacobi matrix is \[ \begin{bmatrix} 2 & -3 \\ 8x & 0 \end{bmatrix} \].

Step 1: Understanding the Function Components

The function \( \mathbf{f}(x, y) \) is a vector function with two component functions. The first component function is \( f_1(x, y) = 2x - 3y \) and the second component function is \( f_2(x, y) = 4x^2 \). Our goal is to find the derivative of each component function with respect to each variable \( x \) and \( y \).

Step 2: Differentiating the First Component

Differentiate \( f_1(x, y) = 2x - 3y \) with respect to \( x \) to get \( \frac{\partial f_1}{\partial x} = 2 \) and with respect to \( y \) to get \( \frac{\partial f_1}{\partial y} = -3 \).

Step 3: Differentiating the Second Component

Differentiate \( f_2(x, y) = 4x^2 \) with respect to \( x \) to get \( \frac{\partial f_2}{\partial x} = 8x \) and with respect to \( y \) to get \( \frac{\partial f_2}{\partial y} = 0 \) because there is no \( y \) in the function.

Step 4: Constructing the Jacobi Matrix

The Jacobi matrix \( J \) is constructed using the partial derivatives found. The matrix is:\[J = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix} = \begin{bmatrix} 2 & -3 \8x & 0 \end{bmatrix}\]

Key Concepts

Partial Derivatives

When working with functions of multiple variables, it is useful to understand how each variable contributes to the function's output. This is where partial derivatives come into play. A partial derivative measures how the function changes as one particular variable is changed, keeping other variables constant.

In the context of our given function \(\mathbf{f}(x, y)=\begin{bmatrix}2x-3y \ 4x^{2}\end{bmatrix}\), we consider each component separately. For each component, we compute a derivative with respect to each variable:

• For the first component \( f_1(x, y) = 2x - 3y \), the partial derivatives are:
  \(\frac{\partial f_1}{\partial x} = 2\) (partial derivative with respect to \(x\))
  \(\frac{\partial f_1}{\partial y} = -3\) (partial derivative with respect to \(y\))
• For the second component \( f_2(x, y) = 4x^2 \), the partial derivatives are:
  \(\frac{\partial f_2}{\partial x} = 8x\)
  \(\frac{\partial f_2}{\partial y} = 0\) (since \(y\) does not appear in \(f_2\))

This understanding is essential for building the Jacobi matrix.

Vector Function

A vector function, like \(\mathbf{f}(x, y)\), consists of multiple component functions, each potentially of multiple variables.

In our example, \(\mathbf{f}(x, y)\) is presented as:
\[\mathbf{f}(x, y)=\begin{bmatrix}2x-3y \ 4x^{2}\end{bmatrix}\]
Each component of the vector function maps inputs \(x\) and \(y\) to a specific output. The function can be visualized as two separate output streams changing in response to variations in the inputs.

• \(f_1(x, y) = 2x - 3y\) generates one output.
• \(f_2(x, y) = 4x^2\) generates another output, showing no change with respect to \(y\).

The vector function as a whole can be interpreted as a flow of information from input space \((x, y)\) to a two-dimensional output space.

Matrix Construction

To analyze how changes in each variable affect the outputs of a vector function, we construct a Jacobi matrix using calculated partial derivatives. This matrix serves as a bridge between the inputs and the outputs.

For our function \(\mathbf{f}(x, y)\), the Jacobi matrix \(J\) is formed by arranging the partial derivatives as follows:\[J = \begin{bmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{bmatrix} = \begin{bmatrix} 2 & -3 \8x & 0 \end{bmatrix}\]
This structured approach offers several advantages:

• The rows correspond to the component functions \(f_1\) and \(f_2\).
• The columns correspond to the variables \(x\) and \(y\).
• The elements of the matrix represent the sensitivity or contribution of each variable to each output component.

Constructing the matrix provides a compact and organized view of all partial derivatives.

Function Differentiation

Differentiation in the context of functions of multiple variables extends the idea of change from single-variable calculus to functions with multiple inputs. This operation explains background activity for each variable individually and how each affects the function's value.

For example, when differentiating component functions such as \(f_1(x, y) = 2x - 3y\), we look at how changes in \(x\) and \(y\) separately influence the outputs:

• \(\frac{\partial f_1}{\partial x} = 2\) tells us \(f_1\) increases by 2 units for every unit increase in \(x\), keeping \(y\) constant.
• \(\frac{\partial f_1}{\partial y} = -3\) indicates \(f_1\) decreases by 3 units for every unit increase in \(y\), keeping \(x\) constant.

The concept is repeated for other component functions.
This method of differentiation focuses on understanding how each component 'moves' or 'responds' to its respective variables, and it is foundational to constructing a Jacobi matrix.