Partial derivative of a total derivative

In summary, the conversation discusses the calculation of a Jacobian, a mathematical concept used in converting integrals between different coordinate systems. The participants give examples and explanations, and mention the chain rule as a useful tool in this process. They also provide a link to a tutorial and ask for further guidance on a specific problem they are trying to solve.
  • #1
halley00
3
0
Hi,

So, in order to calculate a Jacobian, I need to evaluate a partial derivative of a total derivative, i.e.
Let's say I have a function f(x), how do I calculate something like: ∂(df/dx)/∂f?
 
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  • #2
Is this an actual example or something you just thought of?

I can see you using the Jacobian to convert an integral from say XY cartesian coordinates to polar coordinates but not using f(x) by itself.

integral ( f(x,y) dx dy) ---> integral (f(r,phi) Jacobian(r,phi) dr dphi)

So you'd start with x=r * cos(phi) and y=r * sin(phi) and compute the Jacobian
and then you'd convert f(x,y) to f(r,phi)

https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

Here's a short video tutorial on the Jacobian:

 
Last edited:
  • #3
Thank you for your reply. It is an actual example. I have to solve a system of differential equations using Newton-Raphson method, so, let's say I have a function: f1(y1(x1),y2(x1,x2)) = 0, with something like f1 =dy1/dx1 + something else. Then to construct the Jacobian I'll need to evaluate ∂f1/∂y1, which will lead to ∂(dy1/dx1)/∂y1. Any leads?
 
  • #4
halley00 said:
Thank you for your reply. It is an actual example. I have to solve a system of differential equations using Newton-Raphson method, so, let's say I have a function: f1(y1(x1),y2(x1,x2)) = 0, with something like f1 =dy1/dx1 + something else. Then to construct the Jacobian I'll need to evaluate ∂f1/∂y1, which will lead to ∂(dy1/dx1)/∂y1. Any leads?

Remember the chain rule?

∂f1/∂y1 * ∂y1/∂x1 = ∂f1/∂x1
 

What is a partial derivative?

A partial derivative is a mathematical concept that represents the rate of change of a function with respect to one of its input variables, while holding all other variables constant.

What is a total derivative?

A total derivative is a mathematical concept that represents the overall rate of change of a function with respect to all of its input variables.

How is a partial derivative related to a total derivative?

A partial derivative is a component of the total derivative, as it represents the rate of change of the function with respect to one specific input variable.

Why is it important to understand partial derivatives in relation to total derivatives?

Understanding partial derivatives is important because it allows us to analyze the individual components of a function's total derivative, which can provide insights into the behavior of the function and help us make predictions about its future behavior.

How are partial derivatives and total derivatives used in scientific research?

Partial derivatives and total derivatives are commonly used in scientific research, particularly in fields such as physics, engineering, and economics. They can be used to model and analyze complex systems, make predictions, and optimize processes.

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