Partial Derivative/Examples/u - v + 2 w, 2 u + v + 2 w, u - v + w

Theorem
Let:

Then:

Proof
Partial differentiation $y$ gives:

which can be expressed in matrix form as:


 * $\begin {pmatrix} 1 & -1 & 2 \\ 2 & 1 & -2 \\ 1 & -1 & 1 \end {pmatrix} \begin {pmatrix} \dfrac {\partial u} {\partial y} \\ \dfrac {\partial v} {\partial y} \\ \dfrac {\partial w} {\partial y} \end {pmatrix} = \begin {pmatrix} 0 \\ 0 \\ -1 \end {pmatrix}$

Solving by Cramer's Rule:

The solution can be read directly: