Explain what is meant by an exact differential. The three-dimensional vector field $\mathbf{F}$ is defined by

$$\mathbf{F}=\left(e^{x} z^{3}+3 x^{2}\left(e^{y}-e^{z}\right), e^{y}\left(x^{3}-z^{3}\right), 3 z^{2}\left(e^{x}-e^{y}\right)-e^{z} x^{3}\right)$$

Find the most general function that has $\mathbf{F} \cdot \mathbf{d} \mathbf{x}$ as its differential.

Hence show that the line integral

$$\int_{P_{1}}^{P_{2}} \mathbf{F} \cdot \mathbf{d} \mathbf{x}$$

along any path in $\mathbb{R}^{3}$ between points $P_{1}=(0, a, 0)$ and $P_{2}=(b, b, b)$ vanishes for any values of $a$ and $b$.

The two-dimensional vector field $\mathbf{G}$ is defined at all points in $\mathbb{R}^{2}$ except $(0,0)$ by

$$\mathbf{G}=\left(\frac{-y}{x^{2}+y^{2}}, \frac{x}{x^{2}+y^{2}}\right)$$

$(\mathbf{G}$ is not defined at $(0,0)$.) Show that

$$\oint_{C} \mathbf{G} \cdot \mathbf{d} \mathbf{x}=2 \pi$$

for any closed curve $C$ in $\mathbb{R}^{2}$ that goes around $(0,0)$ anticlockwise precisely once without passing through $(0,0)$.