Consider the vector field vec F = langle 3x ^ 2 + 2y ^ 2 + 3z; 4xy + 2y - z 3x - y - z). (i) Explain why vec F is conservative on the domain \mathcal{D} = \mathbb{R} ^ 3 . (ii) Find a potential function g(x, y, z) for vec F (iii) Compute the work done by vec F to move a particle from the point (- 1, 0, 1) to the point (0, 0, 1). (iv) Does this work depend on a path gamma used to get from the pint (- 1, 0, 1) to the point (0, 0, 1) ?

Let's analyze each part of the problem step-by-step:

### (i) Explain why \(\vec{F}\) is conservative on the domain \(\mathcal{D} = \mathbb{R}^3\).

A vector field \(\vec{F}\) is **conservative** if there exists a scalar potential function \(g(x, y, z)\) such that \(\vec{F} = \nabla g\). One equivalent condition for a vector field to be conservative in \(\mathbb{R}^3\) is that its **curl** must be zero everywhere in the domain. So we check if \(\nabla \times \vec{F} = \vec{0}\).

Given:
\[
\vec{F} = \langle 3x^2 + 2y^2 + 3z, 4xy + 2y - z, 3x - y - z \rangle
\]

To compute \(\nabla \times \vec{F}\), we use the definition of the curl:
\[
\nabla \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}, \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)
\]
where \(\vec{F} = \langle F_x, F_y, F_z \rangle\).

#### Calculating Components of the Curl
1. \( \frac{\partial F_z}{\partial y} = -1 \)
2. \( \frac{\partial F_y}{\partial z} = -1 \)

So, the first component of \(\nabla \times \vec{F}\) is:
\[
\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} = -1 - (-1) = 0
\]

3. \( \frac{\partial F_x}{\partial z} = 3 \)
4. \( \frac{\partial F_z}{\partial x} = 3 \)

So, the second component of \(\nabla \times \vec{F}\) is:
\[
\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} = 3 - 3 = 0
\]

5. \( \frac{\partial F_y}{\partial x} = 4y \)
6. \( \frac{\partial F_x}{\partial y} = 4y \)

So, the third component of \(\nabla \times \vec{F}\) is:
\[
\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} = 4y - 4y = 0
\]

Since all components of \(\nabla \times \vec{F}\) are zero, we conclude:
\[
\nabla \times \vec{F} = \vec{0}
\]
Therefore, \(\vec{F}\) is conservative on \(\mathbb{R}^3\).

---

### (ii) Find a potential function \(g(x, y, z)\) for \(\vec{F}\).

Since \(\vec{F}\) is conservative, there exists a potential function \(g(x, y, z)\) such that:
\[
\vec{F} = \nabla g = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, \frac{\partial g}{\partial z} \right)
\]
We are given:
\[
F_x = \frac{\partial g}{\partial x} = 3x^2 + 2y^2 + 3z
\]
\[
F_y = \frac{\partial g}{\partial y} = 4xy + 2y - z
\]
\[
F_z = \frac{\partial g}{\partial z} = 3x - y - z
\]

To find \(g(x, y, z)\), we integrate each component.

#### Integrating with respect to \(x\):
Integrate \(F_x\) with respect to \(x\):
\[
g(x, y, z) = \int (3x^2 + 2y^2 + 3z) \, dx = x^3 + 2xy^2 + 3xz + h(y, z)
\]
where \(h(y, z)\) is a function of \(y\) and \(z\) alone.

#### Differentiating with respect to \(y\):
\[
\frac{\partial g}{\partial y} = 4xy + \frac{\partial h}{\partial y}
\]
Comparing this with \(F_y = 4xy + 2y - z\), we find:
\[
\frac{\partial h}{\partial y} = 2y - z
\]
Integrating with respect to \(y\):
\[
h(y, z) = y^2 - yz + k(z)
\]
where \(k(z)\) is a function of \(z\) alone.

#### Differentiating with respect to \(z\):
\[
\frac{\partial g}{\partial z} = 3x - y - \frac{d}{dz}(yz) - \frac{d}{dz}

Consider the vector field vec F = ⟨3x^2 + 2y^2 + 3z, 4xy + 2y - z, 3x - y - z⟩. (i) Explain why vec F is conservative on the domain D = ℝ^3. (ii) Find a potential function g(x, y, z) for vec F. (iii) Compute the work done by vec F to move a particle from the point (-1, 0, 1) to the point (0, 0, 1). (iv) Does this work depend on the path used to get from (-1, 0, 1) to (0, 0, 1)?

Explore why the vector field F is conservative, how to find its potential function, and calculate the work done to move a particle in this field. This problem demonstrates concepts in vector calculus, including curl, conservative fields, and the Fundamental Theorem of Line Integrals.

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