Problem 32 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The Jacobi matrix is \( J = \begin{bmatrix} 2(x-y) & -2(x-y) \\ \cos(x-y) & -\cos(x-y) \end{bmatrix} \).

Step 1: Understand the Components of the Vector Function

The vector function \( \mathbf{f}(x, y) = \left[ \begin{array}{c} (x-y)^2 \ \sin(x-y) \end{array} \right] \) contains two components. The first component is \((x-y)^2\) and the second component is \(\sin(x-y)\). We need to compute the derivatives of these components with respect to \(x\) and \(y\).

Step 2: Calculate Partial Derivatives of First Component

For the first component \((x-y)^2\):- Partial derivative with respect to \(x\): \(\frac{\partial}{\partial x} ((x-y)^2) = 2(x-y)\).- Partial derivative with respect to \(y\): \(\frac{\partial}{\partial y} ((x-y)^2) = -2(x-y)\).

Step 3: Calculate Partial Derivatives of Second Component

For the second component \(\sin(x-y)\):- Partial derivative with respect to \(x\): \(\frac{\partial}{\partial x} (\sin(x-y)) = \cos(x-y)\).- Partial derivative with respect to \(y\): \(\frac{\partial}{\partial y} (\sin(x-y)) = -\cos(x-y)\).

Step 4: Construct the Jacobi Matrix

Now that we have the partial derivatives, we can construct the Jacobi matrix \( J \) of \( \mathbf{f} \) as follows:\[J = \begin{bmatrix}\frac{\partial}{\partial x} (x-y)^2 & \frac{\partial}{\partial y} (x-y)^2 \\frac{\partial}{\partial x} \sin(x-y) & \frac{\partial}{\partial y} \sin(x-y)\end{bmatrix} = \begin{bmatrix}2(x-y) & -2(x-y) \\cos(x-y) & -\cos(x-y)\end{bmatrix}\]

Key Concepts

Partial Derivatives

Partial derivatives are essential tools in calculus, especially when dealing with functions of multiple variables. They help us understand how a function's value changes as one particular variable changes, while all other variables remain fixed. In the context of the given exercise, we used partial derivatives to analyze each component of a vector function separately.

• To find the partial derivative of a function with respect to a variable, we treat all other variables as constants.
• The calculation involves differentiating the function as you would with a single variable calculus, but only concerning the variable you are deriving by.
• For example, in dealing with the component \((x-y)^2\), taking the partial derivatives with respect to \(x\) and \(y\) showed how the expression changes individually concerning each variable.
• The process also applied to the trigonometric component \(\sin(x-y)\), highlighting the changes in both the \(x\) and \(y\) directions.

Understanding partial derivatives is crucial for constructing the Jacobi matrix, as each element in the matrix is a partial derivative of the vector function components.

Vector Function

A vector function is a function that takes one or more variables and returns a vector. In this exercise, the function \( \mathbf{f}(x, y) = \left[ \begin{array}{c} (x-y)^2 \ \sin(x-y) \end{array} \right] \) indicates that it returns a 2D vector for every permutation of \(x\) and \(y\).

• Each function component represents a separate equation that forms part of the larger vector output.
• Vector functions are used extensively in fields requiring multi-dimensional analysis, such as physics and engineering.
• In our case, each component \((x-y)^2\) and \(\sin(x-y)\) shows different behaviors as \(x\) and \(y\) change, which can be captured using partial derivatives.
• By examining these behaviors, we can develop insights into complex systems by studying the sensitivity and impact of small changes in input variables.

This comprehensive understanding enables us to construct more advanced structures, like the Jacobi matrix, which is vital in analyzing transformations and dynamics of vector fields.

Calculus

Calculus is the mathematical study of change, and it provides tools and techniques to analyze functions concerning variable changes. The Jacobi matrix exercise incorporates fundamental calculus concepts, such as differentiation and partial derivatives, for analyzing the vector function.

• The derivation of the Jacobi matrix itself is an application of these principles, where understanding how a function's output changes with inputs is crucial.
• Differentiation helps in determining slopes and rates of change, which are essential in optimization problems and understanding dynamic systems.
• For multivariable functions, calculus extends these concepts through partial derivatives and matrices, allowing for a more detailed analysis.
• Calculus also provides powerful tools to predict future behavior using derivatives, making it indispensable in scientific and engineering applications.

The practice of calculus through such problems allows students to comprehend not only the mechanics of the subject but its applicability to real-world phenomena and complex mathematical structures like the Jacobi matrix.