Prove jacobian matrix is identity of matrix of order 3

In summary, the Jacobian matrix Df(x,y,z) is the identity matrix of order 3, because the D operator is linear and D1f(x,y,z) = i, D2f(x,y,z) = j, and D3f(x,y,z) = k. This can be represented as Df(x,y,z) = ((1,0,0),(0,1,0),(0,0,1)), where the order of 3 refers to a 3x3 matrix.
  • #1
CrimsnDragn
4
0
If f(x,y,z) = xi + yj +zk, prove that Jacobian matrix Df(x,y,z) is the identity matrix of order 3.

Because the D operator is linear, D1f(x,y,z) = i, D2f(x,y,z) = k, D3f(x,y,z) = k

There is clearly a relationship between this and some sort of identity, but I'm not sure how to state it, and I don't understand the order of linear transformations. Could someone help me?
 
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  • #2
*typo on D2f(x,y,z) = j

actually I was just rethinking about the problem. could Df(x,y,z) = ((1,0,0),(0,1,0),(0,0,1)), which becomes an identity matrix, and the order of 3 refers to 3x3 matrix?
 
  • #3
CrimsnDragn said:
*typo on D2f(x,y,z) = j

actually I was just rethinking about the problem. could Df(x,y,z) = ((1,0,0),(0,1,0),(0,0,1)), which becomes an identity matrix, and the order of 3 refers to 3x3 matrix?

Yes, exactly.
 
  • #4
awesome. thanks!
 

1. What is the Jacobian matrix of a matrix of order 3?

The Jacobian matrix of a matrix of order 3 is a square matrix that represents the partial derivatives of a system of equations with three variables. It is often used in multivariate calculus to solve for the derivatives of a function.

2. How can we prove that a Jacobian matrix is the identity of a matrix of order 3?

To prove that a Jacobian matrix is the identity of a matrix of order 3, we must show that each element in the Jacobian matrix is equal to the corresponding element in the identity matrix. This can be done by taking the partial derivatives of the system of equations and comparing them to the elements in the identity matrix.

3. Why is it important to prove the Jacobian matrix is the identity of a matrix of order 3?

Proving that the Jacobian matrix is the identity of a matrix of order 3 is important in order to ensure the accuracy and validity of calculations involving multivariate functions. It also helps to establish the relationship between the partial derivatives of a function and the elements in the identity matrix.

4. Can the Jacobian matrix be the identity of a matrix of any order?

No, the Jacobian matrix can only be the identity of a matrix of order 3. This is because the Jacobian matrix is defined as a square matrix with the same number of rows and columns as the number of variables in the system of equations, which in this case is three.

5. How is the Jacobian matrix related to the determinant of a matrix of order 3?

The Jacobian matrix is related to the determinant of a matrix of order 3 through the Jacobian determinant, also known as the determinant of the Jacobian matrix. This determinant represents the scaling factor between the original space and the transformed space of a multivariate function.

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