how do i solve the attached calculus problem? For each of the following vector fields, find its curl and determine if it is a gradient field.(a) \(\vec{F} = 2yz \, \hat{i} + (2xz + x^2) \, \hat{j} + (2xy + 2yz) \, \hat{k}\).\[\text{curl } \vec{F} = \boxed{0}\]\(\vec{F}\) is a gradient field. ✅(b) \(\vec{G} = (2xy + yz) \, \hat{i} + (2x^2 + z^2) \, \hat{j} + 5xz \, \hat{k}\).\[\text{curl } \vec{G} = \boxed{-2x \, \hat{i} - 5x \, \hat{j} + (2x - 2y) \, \hat{k}}\]\(\vec{G}\) is not a gradient field. ✅(c) \(\vec{H} = (4xy + 2x^3) \, \hat{i} + (2x^2 + z^2) \, \hat{j} + (2yz - 5z) \, \hat{k}\).\[\text{curl } \vec{H} = \boxed{0}\]\(\vec{H}\) is a gradient field. ✅

Given : F→= 2 yzi→ +2xz+z2j→ +2xy+2yzk→ G→= 2 xy+yzi→ +2x2+z2j→ +5xzk→H→= 4 xy+2x3i→ +2x2+z2j→ +2yz-5zk→ To find: a Curl F⇀b Curl G⇀c Curl H⇀(a) F→= 2 yzi→ +2xz+z2j→ +2xy+2yzk→ Curl F→=i→j→k→∂∂x∂∂y∂∂z2yz2xz+z22xy+2yzCurl F→=∂∂y2xy+2yz-∂∂z2xz+z2i→-∂∂x2xy+2yz-∂∂z2yzj→+∂∂x2xz+z2-∂∂y2yzk→Curl F→=2x+2z-2x-2zi→-2y-2yj→+2z-2zk→Curl F→=0i→-0j→+0k→ F→=∇fF→= 2 yzi→ +2xz+z2j→ +2xy+2yzk→P=2yzQ=2xy+2yz∂P∂y=2z∂Q∂x=2z ∂P∂y=∂Q∂x F→ is gradient field. bG→= 2 xy+yzi→ +2x2+z2j→ +5xzk→Curl G→=i→j→k→∂∂x∂∂y∂∂z2xy+yz2x2+z25xzCurl G→=∂∂y5xz-∂∂z2x2+z2i→-∂∂x5xz-∂∂z2xy+yzj→+∂∂x2x2+z2-∂∂y2xy+yzk→Curl G→=0-2zi→-5z-yj→+4x-2x-zk→Curl G→=-2zi→-5z-yj→+2x-zk→ G→=∇gP=2xy+yzQ=2x2+z2∂P∂y=2x+z∂Q∂x=4x∂P∂y=∂Q∂x G→ is not gradient field.c H→= 4 xy+2x3i→ +2x2+z2j→ +2yz-5zk→ Curl H→=i→j→k→∂∂x∂∂y∂∂z4xy+2x32x2+z22yz-5zCurl H→=∂∂y2yz-5z-∂∂z2xz+z2i→-∂∂x2yz-5z-∂∂z4xy+2x3j→+∂∂x2x2+z2-∂∂y4xy+2x3k→Curl H→=2z-2zi→-0j→+4x-4xk→Curl H→=0i→-0j→+0k→ H→=∇hP=4xy+2x3Q=2x2+z2∂P∂y=4x∂Q∂x=4x H→ is gradient field.