Question

Consider the vector field F=(x2+y2,4xy). Compute the line integrals ∫c1F⋅ds and ∫c2F⋅ds, where c1(t)=(t,t2) and c2(t)=(t,t) for 0≤t≤1. Can you decide from your answers whether or not F is a gradient vector field? Why or why not?

Answer 1

• Integral over $C_1$:

$\displaystyle\int_{C_1}\vec F\cdot\mathrm d\vec s=\int_0^1(t^2+t^4,4t^3)\cdot(1,2t)\,\mathrm dt=\int_0^1(t^2+9t^4)\,\mathrm dt=\boxed{\frac{32}{15}}$

• Integral over $C_2$:

$\displaystyle\int_{C_2}\vec F\cdot\mathrm d\vec s=\int_0^1(2t^2,4t^2)\cdot(1,1)\,\mathrm dt=\int_0^16t^2\,\mathrm dt=\boxed{2}$

The value of the line integral depends on the path, so $\vec F$ is not a gradient vector field.