## Math Problem Statement

Consider the vector field ( \mathbf{F}(x,y,z)= (x^2y + yz^2, x^3+4xz+2,2x + y^2z) ). Find the potential function ( \phi(x,y,z) ) such that ( \mathbf{F} = \nabla \phi ). Assume ( \phi(x,y,z) ) satisfies the condition ( \phi(0,0,0)=5 ). Find the specific potential function that meets this condition.

## Solution

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## Math Problem Analysis

### Mathematical Concepts

Vector Calculus

Potential Function

Gradient Field

### Formulas

Gradient of scalar field: \( \nabla \phi = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \right) \)

### Theorems

Fundamental Theorem of Gradient (Conservative Fields)

### Suitable Grade Level

Undergraduate (Calculus III level)

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