# The Jacobian of a Function

• ver_mathstats
In summary, the conversation revolves around calculating Jacobians with one function and three variables. The suggested approach is to use the function (s^2+sin(rt)-3)/s and substitute values of (1, pi, -1) into the Jacobi matrix, which is equivalent to the gradient in this case. A typo was also mentioned.

#### ver_mathstats

Homework Statement
We must find the Jacobian of f(s,r,t)=s^2+sin(rt)-3. Compute J(f/s)(1, pi, -1).
Relevant Equations
f(s,r,t)=s^2+sin(rt)-3. Compute J(f/s)(1, pi, -1).
I'm used to calculating Jacobians with several functions, so my only question would be how do I approach solving this one with only one function but three variables?

I think our function becomes (s^2+sin(rt)-3)/since we are looking for J(f/s). So then would our Jacobian simply be J=[∂f/∂s ∂f/∂r ∂f/∂t] with finally our values substituted of (1, pi,-1)?

Yes. The Jacobi matrix is simply the gradient in this case.

ver_mathstats
fresh_42 said:
Yes. The Jacobi matrix is simply the gradient in this case.
Thank you, and sorry for the typo, I had meant "I think our function becomes (s^2+sin(rt)-3)/s since we are looking for J(f/s)."

## 1. What is the Jacobian of a function?

The Jacobian of a function is a matrix of partial derivatives that describes the rate of change of a vector-valued function with respect to its input variables. It is used to determine the local behavior of a function near a specific point.

## 2. Why is the Jacobian important in mathematics?

The Jacobian is important in mathematics because it helps to understand the behavior of multivariable functions. It is used in many areas of mathematics, such as optimization, differential equations, and geometry.

## 3. How is the Jacobian related to the gradient?

The Jacobian and the gradient are closely related, as the gradient is the transpose of the Jacobian. This means that the gradient can be thought of as a row vector, while the Jacobian is a column vector.

## 4. Can the Jacobian be used to find the volume of a function?

Yes, the Jacobian can be used to find the volume of a function. This is because the determinant of the Jacobian matrix represents the scaling factor for the volume of a region in multivariable calculus.

## 5. How is the Jacobian used in machine learning?

The Jacobian is used in machine learning to calculate the gradient of a cost function, which is then used to update the parameters of a model through techniques such as gradient descent. It is also used in neural networks to calculate the backpropagation algorithm, which is used to update the weights of the network.