Is there a function whose derivative is the following matrix? If so, find it, if not, explain why not. 1 1 2 2
Is there a function whose derivative is the following matrix? If so, find it, if not, explain why not. 1 1 2 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Question:**
Is there a function whose derivative is the following matrix? If so, find it, if not, explain why not.
\[
\begin{bmatrix}
1 & 1 \\
2 & 2 \\
\end{bmatrix}
\]
**Discussion:**
To determine if there exists a function whose derivative is this matrix, we must consider the properties of differentiable functions and their derivatives.
1. **Derivative Matrix:**
- A derivative matrix, or Jacobian matrix, of a function should satisfy the condition of being integrable, meaning there should exist a potential function that results in this matrix when differentiated.
2. **Criteria for Existence:**
- For a function \( f: \mathbb{R}^n \to \mathbb{R}^m \) to have a derivative \( J \), the mixed partial derivatives must be equal. This is known as the symmetry of second derivatives or Clairaut's theorem (assuming continuity).
3. **Matrix Analysis:**
- Given the matrix:
\[
\begin{bmatrix}
1 & 1 \\
2 & 2 \\
\end{bmatrix}
\]
- The entries of a typical Jacobian are \( \frac{\partial f_1}{\partial x_1}, \frac{\partial f_1}{\partial x_2} \) for the first row and \( \frac{\partial f_2}{\partial x_1}, \frac{\partial f_2}{\partial x_2} \) for the second row.
4. **Check Symmetry:**
- For the given matrix, check if \( \frac{\partial}{\partial x_2} (1) = \frac{\partial}{\partial x_1} (2) \).
- Here, 1 is not equal to 2, which means \( \frac{\partial^2 f_1}{\partial x_2 \partial x_1} \neq \frac{\partial^2 f_2}{\partial x_1 \partial x_2} \).
5. **Conclusion:**
- Since the mixed partial derivatives are not equal, the matrix cannot originate from a single differentiable function. Therefore, there is no function whose derivative is this matrix.
This type of problem helps us understand the importance of the symmetry of mixed partial derivatives](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83b97d9f-eac5-4c59-98f4-44d1d3519004%2Ffe9e5e1a-eefb-4032-b2eb-bb04a2e09468%2Fsp3g76e.png&w=3840&q=75)
Transcribed Image Text:**Question:**
Is there a function whose derivative is the following matrix? If so, find it, if not, explain why not.
\[
\begin{bmatrix}
1 & 1 \\
2 & 2 \\
\end{bmatrix}
\]
**Discussion:**
To determine if there exists a function whose derivative is this matrix, we must consider the properties of differentiable functions and their derivatives.
1. **Derivative Matrix:**
- A derivative matrix, or Jacobian matrix, of a function should satisfy the condition of being integrable, meaning there should exist a potential function that results in this matrix when differentiated.
2. **Criteria for Existence:**
- For a function \( f: \mathbb{R}^n \to \mathbb{R}^m \) to have a derivative \( J \), the mixed partial derivatives must be equal. This is known as the symmetry of second derivatives or Clairaut's theorem (assuming continuity).
3. **Matrix Analysis:**
- Given the matrix:
\[
\begin{bmatrix}
1 & 1 \\
2 & 2 \\
\end{bmatrix}
\]
- The entries of a typical Jacobian are \( \frac{\partial f_1}{\partial x_1}, \frac{\partial f_1}{\partial x_2} \) for the first row and \( \frac{\partial f_2}{\partial x_1}, \frac{\partial f_2}{\partial x_2} \) for the second row.
4. **Check Symmetry:**
- For the given matrix, check if \( \frac{\partial}{\partial x_2} (1) = \frac{\partial}{\partial x_1} (2) \).
- Here, 1 is not equal to 2, which means \( \frac{\partial^2 f_1}{\partial x_2 \partial x_1} \neq \frac{\partial^2 f_2}{\partial x_1 \partial x_2} \).
5. **Conclusion:**
- Since the mixed partial derivatives are not equal, the matrix cannot originate from a single differentiable function. Therefore, there is no function whose derivative is this matrix.
This type of problem helps us understand the importance of the symmetry of mixed partial derivatives
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Yes there exist a function satisfying the .
The matrix is ,
The function
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