Q1: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2^7$. What is the gradient of $f$?

A1: $$ \mathrm{grad}(f)(x_1, x_2) = \begin{pmatrix} 2 x_2^7 \ 14 x_1 x_2^6 \end{pmatrix}$$

A2: $$ \mathrm{grad}(f)(x_1, x_2) = \begin{pmatrix} 2 x_2^7 \ 1 \ 14 x_1 x_2^6 \end{pmatrix}$$

A3: $$ \mathrm{grad}(f)(x_1, x_2) = \begin{pmatrix} 2 x_1^7 \ x_1 x_2^6 \end{pmatrix}$$

A4: $$ \mathrm{grad}(f)(x_1, x_2) = \begin{pmatrix} 2 x_1 x_2\end{pmatrix}$$

Q2: Consider the vector function $v: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $$v(x_1, x_2) = \begin{pmatrix} 2 x_2^7 \ 14 x_1 x_2^6 \end{pmatrix}$$ Does $v$ have a potential function?

A1: Yes, infinitely many ones.

A2: Yes, but only one.

A3: No, there is no potential function.

A4: One needs more information.

Q3: What is the necessary condition for $v: \mathbb{R}^n \rightarrow \mathbb{R}^n$ having a potential function $f \in C^2(\mathbb{R}^n)$?

A1: $v$ is continuously differentiable with $$ \frac{\partial v_i}{\partial x_j}(x) = \frac{\partial v_j}{\partial x_i}(x) $$ for all $x, i, j$.

A2: $v$ is two-times continuously differentiable with $$ \frac{\partial v_i}{\partial x_j}(x) = \frac{\partial v_i}{\partial x_i}(x) $$ for all $x, i, j$.

A3: $v$ is two-times continuously differentiable with $$ \frac{\partial v_j}{\partial x_j}(x) = \frac{\partial v_i}{\partial x_i}(x) $$ for all $x, i, j$.

A4: $v$ is continuously differentiable with $$ \frac{\partial v_i}{\partial x_i}(x) = \frac{\partial v_j}{\partial x_j}(x) $$ for all $x, i, j$.