Consider the vector field\mathbf{F} = (x^2 + y^2, 8 xy). Compute the line integrals
\int_{\mathbf{c}_1} \mathbf{F}\cdot d\mathbf{r}and\int_{\mathbf{c}_2} \mathbf{F}\cdot d\mathbf{r}, where\mathbf{c}_1(t) = (t, t^2)and\mathbf{c}_2(t) = (t, t)for0 \le t \le 1. Can you decide from your answers whether or not
\mathbf{F}
is a gradient vector field? Why or why not?
\int_{\mathbf{c}_1} \mathbf{F}\cdot d\mathbf{r} = ____
\int_{\mathbf{c}_2} \mathbf{F}\cdot d\mathbf{r} = ____
Is\mathbf{F}