Let \(C\) be a smooth curve in\(\mathbb R^3\) parameterized by a vector function \(\mathbf r: [a,b] \to \mathbb R^3\). Define the identity vector field \(\mathbf F(x,y,z) = (x,
y, z)\)
- Show that \[ \int_C \mathbf{F} \cdot \mathrm d\mathbf{r} = \frac{1}{2} \left( \vert\vert \mathbf{r}(b)\vert \vert^2 - \vert \vert \mathbf{r}(a)\vert\vert^2 \right) .\]
- What can you conclude about \(\int_C \mathbf F \cdot \mathrm d\mathbf r\) when \(C\) is a closed curve?