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For a triangular element $T$ with nodes $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, I have the finite element basis functions $\eta_1$, $\eta_2$, and $\eta_3$ as:

$$\eta_j(x_k, y_k) = \delta_{jk}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$

which can be defined by:

$$\eta_j(x, y) = \det\left({\begin{bmatrix} 1& x & y\\ 1& x_{j+1} & y_{j+1}\\ 1& x_{j+2} & y_{j+2}\\ \end{bmatrix}}\right) / \det{\left(\begin{bmatrix} 1& x_j & y_j\\ 1& x_{j+1} & y_{j+1}\\ 1& x_{j+2} & y_{j+2}\\ \end{bmatrix}\right)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$$

I am having trouble proving that:

$$\nabla\eta_j(x, y) = \frac{1}{2\lvert T\lvert}\begin{pmatrix}y_{j+1} - y_{j+2}\\x_{j+2} - x_{j+1}\end{pmatrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$$

where $\lvert T\lvert$ is the area of $T$

$$2\lvert T\lvert = \det\begin{pmatrix}x_2 - x_1 & x_3 - x_1\\y_2 - y_1 & y_3 - y_1\end{pmatrix}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$$

So, when I find the first determinant in equation (2), I have

$$\det\left({\begin{bmatrix} 1& x & y\\ 1& x_{j+1} & y_{j+1}\\ 1& x_{j+2} & y_{j+2}\\ \end{bmatrix}}\right) = (x_{j+1}y_{j+2} - x_{j+2}y_{j+1}) - x(y_{j+2} - y_{j+1}) + y(x_{j+2} - x_{j+1})$$

Why does the term $(x_{j+1}y_{j+2} - x_{j+2}y_{j+1})$ have no derivative with respect to $x$ and $y$ and how does the area of $T$ show up in the equation (3)?

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