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.
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}