Understanding this notation to apply fundamental theorem of calculus Let $F:\mathbb{R}^3\to \mathbb{R}^3$ be of class $C^\infty$. Let $y\in\mathbb{R}^3$ be a nonnull fixed vector. During calculus class, we used this notation:
$$(\partial_{x_1} F \cdot y, \partial_{x_2} F \cdot y, \partial_{x_3} F \cdot y),$$
but it is not definitely clear what does it mean. As $F:\mathbb{R}^3\to \mathbb{R}^3$ I would say that it is the vector given by the product between the rows of the Jacobian matrix and the vector $y$, but I am not sure.
Could someone please clarify that?
Also, it is required to apply the fundamental theorem of calculus to $(\partial_{x_1} F \cdot y, \partial_{x_2} F \cdot y, \partial_{x_3} F \cdot y)$. To be more precise, defined
$$G_y:\mathbb{R}^3\to \mathbb{R}^3 \quad\text{such that}\quad x\in\mathbb{R}^3\mapsto G_y(x) = (\partial_{x_1} F \cdot y, \partial_{x_2} F \cdot y, \partial_{x_3} F \cdot y),$$
I have to evaluate $G_y(x) - G_y(0)$ by using the fundamental theorem of calculus. I would write that
$$G_y(x) - G_y(0) = \int_0^1 G_y^{\prime}(sx) ds,$$
but I don't know what $G^{\prime}$ is (the notation $G^{\prime}$ maybe it is not appropriate, but it is to identify that I have to put a derivative inside the integral).
Could someone please help in understanding that too?
Thank you in advance.
 A: Usually, for $F: \mathbb R^3 \to \mathbb R^3$, the gradient is the matrix $n \times n$ given by
$$\nabla F = 
\begin{pmatrix}
\nabla F_1\\
\nabla F_2\\
\nabla F_3
\end{pmatrix} = \begin{pmatrix}
\partial_1 F_1 & \partial_2 F_1 & \partial_3 F_1 \\
\partial_1 F_2 & \partial_2 F_2 & \partial_3 F_2 \\
\partial_1 F_3 & \partial_2 F_3 & \partial_3 F_3 
\end{pmatrix} = \begin{pmatrix}
\partial_1 F & \partial_2 F & \partial_3 F 
\end{pmatrix}.$$
In your case, for $y \in \mathbb R^3$,
$$\nabla F\cdot y = \begin{pmatrix}
\partial_1 F \cdot y& \partial_2 F\cdot y & \partial_3 F\cdot y \\
\end{pmatrix} = \begin{pmatrix}
\partial_1 F_1 & \partial_2 F_1 & \partial_3 F_1 \\
\partial_1 F_2 & \partial_2 F_2 & \partial_3 F_2 \\
\partial_1 F_3 & \partial_2 F_3 & \partial_3 F_3 
\end{pmatrix}\begin{pmatrix}
y_1\\
y_2\\
y_3
\end{pmatrix} \in \mathbb R^3.$$
The generalization of the fundamental theorem of calculus for $f: \mathbb R^n \to \mathbb R$ is
$$f(r(a)) - f(r(b)) = \int_a^b \nabla f(r(t)) \cdot r'(t) dt.$$
Now as $G$ values in $\mathbb R^3$, you must apply this formula componentwise, i.e. for $i \in \{1, 2, 3\}$,
\begin{align}
G_y^i(x) - G_y^i(0) &= \int_0^1 \nabla\left(\sum_j\partial_j F_i y_j\right)(sx_1, sx_2, sx_3)\begin{pmatrix}
x_1\\
x_2\\
x_3
\end{pmatrix} ds\\
&= \int_0^1 \left(\sum_{jk}\partial_{jk}^2 F_i y_jx_k\right)(sx_1, sx_2, sx_3)ds
\end{align}
