Question

Berechnen Sie die Ableitung der Verkettung \((\mathrm{f} \circ \mathbf{g})\) der Abbildungen $$ \mathbf{g}(x, y)=\left(\sin x, \cos y, e^{x y}\right)^{T} \quad \text { und } \quad \mathbf{f}\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}, x_{1}^{2}, x_{2}^{2}, x_{3}^{2}\right)^{T} $$

Short answer

Calculate derivatives of compositions using chain rule and matrix multiplication.

Step-by-step solution

Understanding the Composition

The functions given are \( \mathbf{g}(x, y)=\left(\sin x, \cos y, e^{xy}\right)^T \) and \( \mathbf{f}(x_1, x_2, x_3)=\left(x_1, x_1^2, x_2^2, x_3^2\right)^T \). We are asked to find the derivative of the composition \((\mathrm{f} \circ \mathbf{g})(x, y)\). This means we first apply \( \mathbf{g}(x, y) \) and then \( \mathbf{f} \) to the result of \( \mathbf{g} \).

Computing the Intermediate Function

First, calculate \( \mathbf{g}(x, y) \): this gives the vector \( (\sin x, \cos y, e^{xy})^T \). Then substitute these into \( \mathbf{f} \): \( \mathbf{f}\left(g_1(x,y), g_2(x,y), g_3(x,y)\right) = \left(\sin x, (\sin x)^2, (\cos y)^2, (e^{xy})^2\right)^T \).

Apply the Chain Rule for Derivatives

The derivative of the composition \( (\mathrm{f} \circ \mathbf{g})(x, y) \) is found using the chain rule. This requires the Jacobian matrices of \( \mathbf{f} \) and \( \mathbf{g} \). The Jacobian \( J_g(x,y) \) is computed as \( \begin{bmatrix} \cos x & 0 \ 0 & -\sin y \ y e^{xy} & x e^{xy} \end{bmatrix} \). Then compute \( J_f(x_1, x_2, x_3) \) as \( \begin{bmatrix} 1 & 0 & 0 \ 2x_1 & 0 & 0 \ 0 & 2x_2 & 0 \ 0 & 0 & 2x_3 \end{bmatrix} \).

Combine Jacobians for the Composition

Using the chain rule for the derivative of a composition, we compute \( (J_f \circ \mathbf{g})(x, y) \cdot J_g(x, y) \). This necessitates evaluating \( J_f \) at \( \mathbf{g}(x, y) = (\sin x, \cos y, e^{xy}) \). We get \( J_f(\sin x, \cos y, e^{xy}) = \begin{bmatrix} 1 & 0 & 0 \ 2\sin x & 0 & 0 \ 0 & 2\cos y & 0 \ 0 & 0 & 2e^{xy} \end{bmatrix} \). Multiply this matrix by \( J_g \) to obtain the derivative of the composition.

Calculate the Matrix Product

Perform the matrix multiplication: \[\begin{bmatrix} 1 & 0 & 0 \2\sin x & 0 & 0 \0 & 2\cos y & 0 \0 & 0 & 2e^{xy} \end{bmatrix} \begin{bmatrix} \cos x & 0 \0 & -\sin y \y e^{xy} & x e^{xy} \end{bmatrix}\]Compute each element of the resulting matrix to get the full derivative matrix \( d(\mathrm{f} \circ \mathbf{g}) \).

Key concepts

Jacobian Matrix

The Jacobian matrix is a pivotal concept in multivariable calculus. It essentially captures how a vector-valued function changes as input variables change. For any given function mapping from \mathbb{R}^n\ to \mathbb{R}^m\, the Jacobian is a matrix of all first-order partial derivatives of the function's vector components with respect to its input variables.

When you're dealing with functions like \( \mathbf{g}(x, y)=(\sin x, \cos y, e^{xy})^T \) and \( \mathbf{f}(x_1, x_2, x_3)=(x_1, x_1^2, x_2^2, x_3^2)^T \), the Jacobian matrix is crucial for understanding and computing their derivatives.

In our specific example, we need to derive the Jacobian of each function separately.
For \( \mathbf{g}(x, y) \), the Jacobian matrix is given by:

\[ J_g(x,y) = \begin{bmatrix} \cos x & 0 \ 0 & -\sin y \ y e^{xy} & x e^{xy} \end{bmatrix} \]

This matrix captures how each component of \( \mathbf{g} \) changes with respect to \( x \) and \( y \). Similarly, the Jacobian for \( \mathbf{f}(x_1, x_2, x_3) \) is:

\[ J_f(x_1, x_2, x_3) = \begin{bmatrix} 1 & 0 & 0 \ 2x_1 & 0 & 0 \ 0 & 2x_2 & 0 \ 0 & 0 & 2x_3 \end{bmatrix}. \]

Understanding the composition of these Jacobians is vital for utilizing the chain rule effectively.

Function Composition

Function composition is a powerful concept in mathematics where you apply one function to the results of another function. In the context of derivatives, understanding function composition helps determine how changes in input influence the composition's output.

In our exercise, we are asked to find the derivative of the composition \((\mathrm{f} \circ \mathbf{g})(x, y)\). This essentially means that we first apply \(\mathbf{g}\) on inputs \(x\) and \(y\) to get a resulting vector, and then apply \(\mathbf{f}\) on this resulting vector.

The process of composition can be thought of as:

• First, substitute the output of \( \mathbf{g} \) directly into \( \mathbf{f} \) as its input variables.
• This forms the chain of functions, thus requiring the chain rule to find the derivative of the composite function.

For example, when applying \( \mathbf{g}(x, y) = (\sin x, \cos y, e^{xy})^T \), it gives you inputs for \( \mathbf{f} \), which then becomes \( \mathbf{f}(\sin x, \cos y, e^{xy}) = (\sin x, (\sin x)^2, (\cos y)^2, (e^{xy})^2)^T \).

This process highlights why derivatives of compositions involve multiple steps and use of Jacobians to handle multiple variables and their interactions.

Partial Derivatives

Partial derivatives are the foundation for understanding the changes in functions with multiple variables. They are crucial in calculating Jacobian matrices, especially when dealing with vector-valued functions.

When given a function like \( \mathbf{g}(x, y) = (\sin x, \cos y, e^{xy})^T \), partial derivatives allow us to compute how each component of the function changes with respect to individual input variables, \(x\) and \(y\).

In our context, finding partial derivatives means:

• The partial derivative of \( \sin x \) with respect to \(x\) is \( \cos x \), capturing the sensitivity of \( \sin x \) regarding \(x\).
• The partial derivative of \( \cos y \) with respect to \(y\) is \(-\sin y \), reflecting its change concerning \(y\).
• For \( e^{xy} \), the partial derivatives involve product rule: with respect to \(x\), it becomes \( y e^{xy} \) and with respect to \(y\), it's \( x e^{xy} \).

These derivatives are lined up neatly in the form of a Jacobian matrix.

Similarly, for the function \( \mathbf{f}(x_1, x_2, x_3) = (x_1, x_1^2, x_2^2, x_3^2)^T \), partial derivatives dictate how the output scales with each respective variable input, forming part of the extended derivative through the chain rule for compositions.

Mastering partial derivatives equips students with the fundamental tools needed for more advanced calculus, especially when dealing with multidimensional systems.