Find a Function F Such That $\mathbf { F } = \nabla f$

Short Answer

Find a function f such that $\mathbf { F } = \nabla f$ , and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the given curve C. $\mathbf { F } ( x , y ) = e ^ { 6 y } \mathbf { i } + \left( 1 + 6 x e ^ { 6 y } \right) \mathbf { j } , \quad C : \mathbf { r } ( t ) = t e ^ { t } \mathbf { i } + ( 1 + t ) \mathbf { j } , 0 \leq t \leq 1$