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

---

Definitions:

Agency Relationship

A relationship between two parties in which one party (the agent) agrees to represent or act for another party (the principal).

Restatement (Third)

A reference to the third series of the Restatements of the Law, which are authoritative summaries of common law in the United States, aimed at clarifying, modernizing, and improving the law.

Automatic Termination

A clause within a contract or agreement that causes it to end automatically upon the occurrence of specific conditions.

Illegal Act

An action that is forbidden by law or statute.