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

question 151

Essay

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 curve C. $\mathbf { F } ( x , y , z ) = 4 x e ^ { x } \mathbf { i } + \cos y \mathbf { j } + 2 x ^ { 2 } \mathrm { e } ^ { z } \mathbf { k } ; C : \mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + t ^ { 4 } \mathbf { k } , \quad 0 \leq t \leq 1$

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