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.

We are given the vector field \(\mathbf{F}(x,y,z) = (x^2y + yz^2, x^3 + 4xz + 2, 2x + y^2z)\), and our goal is to find the potential function \(\phi(x,y,z)\) such that:

\[
\mathbf{F} = \nabla \phi.
\]

This means that \(\mathbf{F}\) is the gradient of \(\phi\), or:

\[
\mathbf{F} = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \right).
\]

This gives the following system of partial differential equations:

\[
\frac{\partial \phi}{\partial x} = x^2y + yz^2, \quad \frac{\partial \phi}{\partial y} = x^3 + 4xz + 2, \quad \frac{\partial \phi}{\partial z} = 2x + y^2z.
\]

### Step 1: Integrating with respect to \(x\)
First, we integrate the first equation with respect to \(x\):

\[
\frac{\partial \phi}{\partial x} = x^2y + yz^2.
\]

Integrating term by term:

\[
\phi(x, y, z) = \int (x^2y + yz^2) \, dx = \frac{x^3}{3}y + xyz^2 + g(y, z),
\]

where \(g(y, z)\) is a function of \(y\) and \(z\) (because the derivative with respect to \(x\) eliminates terms independent of \(x\)).

### Step 2: Differentiating with respect to \(y\)
Next, we differentiate this expression with respect to \(y\) and compare with the second equation:

\[
\frac{\partial \phi}{\partial y} = \frac{\partial}{\partial y} \left( \frac{x^3}{3}y + xyz^2 + g(y, z) \right) = \frac{x^3}{3} + xz^2 + \frac{\partial g(y, z)}{\partial y}.
\]

From the second equation, we know:

\[
\frac{\partial \phi}{\partial y} = x^3 + 4xz + 2.
\]

Equating both expressions for \(\frac{\partial \phi}{\partial y}\):

\[
\frac{x^3}{3} + xz^2 + \frac{\partial g(y, z)}{\partial y} = x^3 + 4xz + 2.
\]

Simplifying this:

\[
\frac{2x^3}{3} + xz^2 + \frac{\partial g(y, z)}{\partial y} = 4xz + 2.
\]

This gives:

\[
\frac{\partial g(y, z)}{\partial y} = 4xz - xz^2 - \frac{2x^3}{3} + 2.
\]

However, upon closer inspection of constants,

Explore how to find the potential function \( \phi(x,y,z) \) for the vector field \( \mathbf{F}(x,y,z) = (x^2y + yz^2, x^3 + 4xz + 2, 2x + y^2z) \), satisfying \( \phi(0,0,0)=5 \). This problem involves vector calculus concepts such as gradient fields and conservative fields, ideal for students in advanced calculus or vector analysis.

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