- #1

andlook

- 33

- 0

[tex]

\mbox{f} : \mathbb{R}^m \mbox{ to } \mathbb{R}^n [/tex]

Then df is what? the jacobian matrix of partial derivatives?

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter andlook
- Start date

In summary, df represents the linear transformation that approximates the map f in differential geometry.

- #1

andlook

- 33

- 0

[tex]

\mbox{f} : \mathbb{R}^m \mbox{ to } \mathbb{R}^n [/tex]

Then df is what? the jacobian matrix of partial derivatives?

Mathematics news on Phys.org

- #2

quasar987

Science Advisor

Homework Helper

Gold Member

- 4,807

- 32

More generally, suppose the map

[tex]f:U\rightarrow\mathbb{R}^n[/tex]

In differential geometry, we define on each point

[tex]

TU:=\bigcup_{x\in U}T_xU

[/tex]

Then

- #3

HallsofIvy

Science Advisor

Homework Helper

- 42,988

- 981

Strictly speaking, df is the linear transformation that best approximates f (in the same way that y= mx+ b best approximates f(x) at x= a when m= f'(a)). Given the standard bases for Randlook said:

[tex]

\mbox{f} : \mathbb{R}^m \mbox{ to } \mathbb{R}^n [/tex]

Then df is what? the jacobian matrix of partial derivatives?

The jacobian matrix of partial derivatives is a mathematical tool used to represent the relationship between multiple variables in a system. It is commonly used in multivariate calculus and is especially useful in solving systems of differential equations.

The jacobian matrix is calculated by taking the partial derivatives of each variable in a system with respect to all other variables and arranging them in a matrix. The result is a square matrix where each entry represents the sensitivity of one variable to changes in another.

The determinant of the jacobian matrix is a measure of how much the variables in a system are changing with respect to each other. It can help determine the stability of a system and the behavior of its solutions over time.

In machine learning, the jacobian matrix is used to calculate the gradient of a multivariate function. This gradient is used in optimization algorithms to find the optimal values for the parameters of a model.

One limitation of the jacobian matrix is that it assumes all variables in a system are continuous and differentiable. It may also become computationally intensive for large systems with many variables. Additionally, the jacobian matrix may not accurately represent the behavior of a system if there are significant nonlinearities present.

- Replies
- 11

- Views
- 2K

- Replies
- 9

- Views
- 1K

- Replies
- 8

- Views
- 2K

- Replies
- 2

- Views
- 2K

- Replies
- 2

- Views
- 695

- Replies
- 1

- Views
- 562

- Replies
- 6

- Views
- 2K

- Replies
- 1

- Views
- 2K

- Replies
- 1

- Views
- 2K

- Replies
- 1

- Views
- 1K

Share: