Computing Derivative Of J_F^{-1}(x)F(x)

Introduction

In multivariable calculus, the Jacobian matrix plays a crucial role in understanding the behavior of functions. Given a $C^2$ function $F:\mathbb R^n\to\mathbb R^n$, the Jacobian matrix $J_F(x)$ at a point $x\in\mathbb R^n$ is a square matrix whose entries are the partial derivatives of the components of $F$. In this article, we will explore the computation of the derivative of $J_F^{-1}(x)F(x)$, where $J_F^{-1}(x)$ is the inverse of the Jacobian matrix.

Jacobian Matrix

The Jacobian matrix $J_F(x)$ of a function $F:\mathbb R^n\to\mathbb R^n$ at a point $x\in\mathbb R^n$ is defined as:

$$J_F(x) = \begin{bmatrix} \frac{\partial F_1}{\partial x_1} & \frac{\partial F_1}{\partial x_2} & \cdots & \frac{\partial F_1}{\partial x_n} \\ \frac{\partial F_2}{\partial x_1} & \frac{\partial F_2}{\partial x_2} & \cdots & \frac{\partial F_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial F_n}{\partial x_1} & \frac{\partial F_n}{\partial x_2} & \cdots & \frac{\partial F_n}{\partial x_n} \end{bmatrix}$$

where $F_i$ is the $i$-th component of the function $F$.

Inverse Jacobian Matrix

The inverse Jacobian matrix $J_F^{-1}(x)$ is defined as:

$$J_F^{-1}(x) = (J_F(x))^{-1}$$

Computing Derivative of J_F^{-1}(x)F(x)

To compute the derivative of $J_F^{-1}(x)F(x)$, we can use the chain rule and the product rule.

Let $G(x) = J_F^{-1}(x)F(x)$. Then, we can write:

$$\frac{\partial G}{\partial x_i} = \frac{\partial}{\partial x_i} (J_F^{-1}(x)F(x))$$

Using the product rule, we can expand the derivative as:

$$\frac{\partial G}{\partial x_i} = \frac{\partial J_F^{-1}(x)}{\partial x_i} F(x) + J_F^{-1}(x) \frac{\partial F(x)}{\partial x_i}$$

Now, we need to compute the derivative of the inverse Jacobian matrix.

Derivative of Inverse Jacobian Matrix

To compute the derivative of the inverse Jacobian matrix, we can use the formula:

$$\frac{\partial J_F^{-1}(x)}{\partial x_i} = -J_F^{-1}(x) \frac{\partial J_F(x)}{\partial x_i} J_F^{-1}(x)$$

This formula can be derived by differentiating the identity $J_F(x) J_F^{-1}(x) = I$ with respect to $x_i$.

Computing the Derivative

Now that we have the formula for the derivative of the inverse Jacobian matrix, we can compute the derivative of $G(x) = J_F^{-1}(x)F(x)$.

$$\frac{\partial G}{\partial x_i} = -J_F^{-1}(x) \frac{\partial J_F(x)}{\partial x_i} J_F^{-1}(x) F(x) + J_F^{-1}(x) \frac{\partial F(x)}{\partial x_i}$$

This is the final formula for the derivative of $J_F^{-1}(x)F(x)$.

Conclusion

In this article, we have explored the computation of the derivative of $J_F^{-1}(x)F(x)$, where $J_F^{-1}(x)$ is the inverse of the Jacobian matrix. We have derived the formula for the derivative of the inverse Jacobian matrix and used it to compute the derivative of $G(x) = J_F^{-1}(x)F(x)$. This formula can be used to study the behavior of functions in multivariable calculus.

References

• [1] Multivariable Calculus by James Stewart
• [2] Vector Analysis by Murray R. Spiegel
• [3] Jacobian by Wikipedia
• [4] Frechet Derivative by Wikipedia

Further Reading

For further reading on multivariable calculus, vector analysis, and the Jacobian matrix, we recommend the following resources:

• Multivariable Calculus by James Stewart
• Vector Analysis by Murray R. Spiegel
• Jacobian by Wikipedia
• Frechet Derivative by Wikipedia

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Introduction

In our previous article, we explored the computation of the derivative of $J_F^{-1}(x)F(x)$, where $J_F^{-1}(x)$ is the inverse of the Jacobian matrix. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the Jacobian matrix?

A: The Jacobian matrix $J_F(x)$ of a function $F:\mathbb R^n\to\mathbb R^n$ at a point $x\in\mathbb R^n$ is a square matrix whose entries are the partial derivatives of the components of $F$.

Q: How do I compute the Jacobian matrix?

A: To compute the Jacobian matrix, you need to find the partial derivatives of the components of the function $F$ with respect to each variable. The Jacobian matrix is then given by:

$$J_F(x) = \begin{bmatrix} \frac{\partial F_1}{\partial x_1} & \frac{\partial F_1}{\partial x_2} & \cdots & \frac{\partial F_1}{\partial x_n} \\ \frac{\partial F_2}{\partial x_1} & \frac{\partial F_2}{\partial x_2} & \cdots & \frac{\partial F_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial F_n}{\partial x_1} & \frac{\partial F_n}{\partial x_2} & \cdots & \frac{\partial F_n}{\partial x_n} \end{bmatrix}$$

Q: What is the inverse Jacobian matrix?

A: The inverse Jacobian matrix $J_F^{-1}(x)$ is defined as:

$$J_F^{-1}(x) = (J_F(x))^{-1}$$

Q: How do I compute the derivative of the inverse Jacobian matrix?

A: To compute the derivative of the inverse Jacobian matrix, you can use the formula:

$$\frac{\partial J_F^{-1}(x)}{\partial x_i} = -J_F^{-1}(x) \frac{\partial J_F(x)}{\partial x_i} J_F^{-1}(x)$$

Q: How do I compute the derivative of J_F^{-1}(x)F(x)?

A: To compute the derivative of $J_F^{-1}(x)F(x)$, you can use the chain rule and the product rule. The derivative is given by:

$$\frac{\partial G}{\partial x_i} = \frac{\partial}{\partial x_i} (J_F^{-1}(x)F(x))$$

$$\frac{\partial G}{\partial x_i} = \frac{\partial J_F^{-1}(x)}{\partial x_i} F(x) + J_F^{-1}(x) \frac{\partial F(x)}{\partial x_i}$$

Q: What is the final formula for the derivative of J_F^{-1}(x)F(x)?

A: The final formula for the derivative of $J_F^{-1}(x)F(x)$ is:

$$\frac{\partial G}{\partial x_i} = -J_F^{-1}(x) \frac{\partial J_F(x)}{\partial x_i} J_F^{-1}(x) F(x) + J_F^{-1}(x) \frac{\partial F(x)}{\partial x_i}$$

Conclusion

In this article, we have answered some frequently asked questions related to the computation of the derivative of $J_F^{-1}(x)F(x)$. We hope this article has been helpful in understanding this topic. If you have any further questions or need further clarification, please don't hesitate to contact us.

References

• [1] Multivariable Calculus by James Stewart
• [2] Vector Analysis by Murray R. Spiegel
• [3] Jacobian by Wikipedia
• [4] Frechet Derivative by Wikipedia

Further Reading

For further reading on multivariable calculus, vector analysis, and the Jacobian matrix, we recommend the following resources:

• Multivariable Calculus by James Stewart
• Vector Analysis by Murray R. Spiegel
• Jacobian by Wikipedia
• Frechet Derivative by Wikipedia

We hope this article has been helpful in understanding the computation of the derivative of $J_F^{-1}(x)F(x)$. If you have any questions or need further clarification, please don't hesitate to contact us.