Problem 1 - Advanced Calculus

Question

Define \(\mathbf{F}(x, y)=\left(e^{x y}+2 x, y^{2}+\sin (x-y)\right) \quad\) for \((x, y)\) in \(\mathbb{R}^{2}\) Find the derivative matrix of the mapping \(\mathbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) at the points (0,0) and \((\pi, 0)\)

Step-by-Step Solution

Answer

The derivative matrix of the given mapping \({\mathbf{F}}\) at (0,0) is \begin{bmatrix}2 & 0 \1 & -1\end{bmatrix}, and at \( (\pi, 0) \) is \(\begin{bmatrix}2 & \pi \-1 & 1\end{bmatrix}\).

Step 1: Understanding the Derivative Matrix

The derivative matrix, also known as the Jacobian matrix, of a mapping \(\mathbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) is a matrix that represents all the first-order partial derivatives of \(\mathbf{F}\). For a function \(\mathbf{F}(x, y) = (f(x, y), g(x, y))\), the derivative matrix is given by \([J_{\mathbf{F}}(x, y)]=\begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{bmatrix}\).

Step 2: Compute the Partial Derivatives

For the given function \(\mathbf{F}(x, y)=(e^{x y}+2x, y^{2}+\sin(x-y))\), calculate the partial derivatives: \(\frac{\partial }{\partial x}(e^{x y}+2x) = y e^{x y} + 2,\, \frac{\partial }{\partial y}(e^{x y}+2x) = x e^{x y},\) \(\frac{\partial }{\partial x}(y^{2}+\sin(x-y)) = \cos(x-y),\, \frac{\partial }{\partial y}(y^{2}+\sin(x-y)) = 2y - \cos(x-y)\).

Step 3: Evaluate the Partial Derivatives at (0, 0)

Substitute the point (0,0) into the partial derivatives to determine their values: \(\frac{\partial }{\partial x}(e^{x y}+2x)\Big|_{(0,0)} = 0 \cdot e^{0 \cdot 0} + 2 = 2,\, \frac{\partial }{\partial y}(e^{x y}+2x)\Big|_{(0,0)} = 0 \cdot e^{0 \cdot 0} = 0,\) \(\frac{\partial }{\partial x}(y^{2}+\sin(x-y))\Big|_{(0,0)} = \cos(0-0) = 1,\, \frac{\partial }{\partial y}(y^{2}+\sin(x-y))\Big|_{(0,0)} = 2 \cdot 0 - \cos(0-0) = -1\). Thus, the derivative matrix at (0,0) is \([J_{\mathbf{F}}(0, 0)]=\begin{bmatrix} 2 & 0 \ 1 & -1 \end{bmatrix}\).

Step 4: Evaluate the Partial Derivatives at \( (\pi, 0) \)

Substitute the point \( (\pi, 0) \) into the partial derivatives to determine their values: \(\frac{\partial }{\partial x}(e^{x y}+2x)\Big|_{(\pi,0)} = 0 \cdot e^{\pi \cdot 0} + 2 = 2,\, \frac{\partial }{\partial y}(e^{x y}+2x)\Big|_{(\pi,0)} = \pi \cdot e^{\pi \cdot 0} = \pi,\) \(\frac{\partial }{\partial x}(y^{2}+\sin(x-y))\Big|_{(\pi,0)} = \cos(\pi-0) = -1,\, \frac{\partial }{\partial y}(y^{2}+\sin(x-y))\Big|_{(\pi,0)} = 0 - \cos(\pi-0) = 1\). Thus, the derivative matrix at \( (\pi, 0) \) is \([J_{\mathbf{F}}(\pi, 0)]=\begin{bmatrix} 2 & \pi \ -1 & 1 \end{bmatrix}\).

Key Concepts

Jacobian Matrix

The Jacobian matrix is a powerful tool in multivariable calculus, representing the gradient of a vector-valued function. Specifically, for a mapping \( \textbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \), it encapsulates how small changes in the input variables cause changes in the output variables.

By arranging the first-order partial derivatives into a matrix, the Jacobian provides a concise way to visualize and compute differential properties of the mapping. It is substantial in fields like engineering and physics, where it aids in understanding the behavior of complex systems.

Partial Derivatives

Partial derivatives represent the rate at which a multivariable function changes as one variable moves infinitesimally while others are held constant. They are fundamental to the study of multivariable calculus because they give insight into the function's behavior along each axis separately.

When calculating the Jacobian matrix, the partial derivatives form its core, giving us a snapshot of the function's local linear approximation. This microscopic viewpoint allows for a granular analysis of the function's behavior around a specific point.

Multivariable Calculus

Multivariable calculus extends the principles of calculus to functions of several variables. Unlike single-variable calculus, it involves more complex entities like scalar fields, vector fields, and vector-valued functions. It allows us to explore high-dimensional spaces and understand phenomena that depend on multiple factors.

Concepts like derivatives and integrals take on new meanings, leading to tools like the gradient, divergence, curl, and the aforementioned Jacobian matrix. These help in describing the geometry and behavior of multivariable functions, making multivariable calculus indispensable in disciplines ranging from economics to meteorology.

Mapping \( \textbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \)

In the context of the exercise, the mapping \( \textbf{F}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \) refers to a transformation from a two-dimensional space onto itself. Each point \( (x, y) \) in the domain is mapped to a new point, defined by the function \( \textbf{F} \).

The mapping can be visual or algorithmic, and interpreting such mappings is a fundamental part of multivariable calculus. Studying how different inputs are transformed under \( \textbf{F} \) unveils the nature of a system described by the mapping, hence the importance of the derivative matrix to represent the mapping's local behavior.