The Jacobian matrix of f(x,y,z)=(x+y+z , xyz , xy+yz+zx) at (1,0,1) is:
J = | 1 1 1 |
| 0 1 0 |
| 1 1 1 |
To find the total derivative of f in the form of linear transformation, we can use the Jacobian matrix as follows:
Let Δx, Δy, and Δz be small changes in x, y, and z, respectively. Then the total derivative of f can be represented as:
Δf = J * Δv
Where Δv = | Δx |
| Δy |
| Δz |
Therefore, the total derivative of f in the form of linear transformation is given by:
Δf = | 1 1 1 | | Δx | = | Δx + Δy + Δz |
| 0 1 0 | | Δy | | Δy |
| 1 1 1 | | Δz | | Δx + Δy + Δz |