For what value of k is →F(x,y,z) = (12xy^2z + 3y)i^ + (12x^2yz + 3x)j^ + k2x Conservative

To determine the value of \( k \) that makes the vector field \(\vec{F}(x,y,z) = (12xy^2z + 3y)\mathbf{i} + (12x^2yz + 3x)\mathbf{j} + k2x\mathbf{k}\) conservative, we need to ensure that the curl of \(\vec{F}\) is zero. A vector field \(\vec{F}\) is conservative if and only if its curl \(\nabla \times \vec{F}\) is zero.

The curl of \(\vec{F} = (F_1, F_2, F_3)\) is given by:
\[
\nabla \times \vec{F} = \left( \frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z} \right)\mathbf{i} + \left( \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x} \right)\mathbf{j} + \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right)\mathbf{k}
\]

For the given vector field:
\[
\vec{F}(x,y,z) = (12xy^2z + 3y)\mathbf{i} + (12x^2yz + 3x)\mathbf{j} + k2x\mathbf{k}
\]

First, we identify the components:
\[
F_1 = 12xy^2z + 3y
\]
\[
F_2 = 12x^2yz + 3x
\]
\[
F_3 = k2x
\]

Now, we compute the partial derivatives needed for the curl:

1. \(\frac{\partial F_3}{\partial y} = \frac{\partial (k2x)}{\partial y} = 0\)

2. \(\frac{\partial F_2}{\partial z} = \frac{\partial (12x^2yz + 3x)}{\partial z} = 12x^2y\)

3. \(\frac{\partial F_1}{\partial z} = \frac{\partial (12xy^2z + 3y)}{\partial z} = 12xy^2\)

4. \(\frac{\partial F_3}{\partial x} = \frac{\partial (k2x)}{\partial x} = k2\)

5. \(\frac{\partial F_2}{\partial x} = \frac{\partial (12x^2yz + 3x)}{\partial x} = 24xyz + 3\)

6. \(\frac{\partial F_1}{\partial y} = \frac{\partial (12xy^2z + 3y)}{\partial y} = 24xyz + 3\)

Next, we substitute these partial derivatives into the curl formula:

\[
\nabla \times \vec{F} = \left(0 - 12x^2y\right)\mathbf{i} + \left(12xy^2 - k2\right)\mathbf{j} + \left(24xyz + 3 - (24xyz + 3)\right)\mathbf{k}
\]

Simplifying this, we get:

\[
\nabla \times \vec{F} = -12x^2y\mathbf{i} + (12xy^2 - k2)\mathbf{j} + 0\mathbf{k}
\]

For \(\nabla \times \vec{F}\) to be zero, each component must be zero:

1. \( -12x^2y = 0 \) implies that this is always zero.
2. \( 12xy^2 - k2 = 0 \) implies \( k2 = 12xy^2 \). For this to hold for all \(x\) and \(y\), \( k \) must be 6.

Therefore, \( k = 6 \) makes the vector field conservative.

Would you like more details on any part of this solution or have any questions?

Here are 5 more questions you might find interesting:

1. How do you determine if a vector field is conservative?
2. What is the significance of the curl of a vector field?
3. Can you show the steps to find a potential function for a conservative vector field?
4. How does the Fundamental Theorem for Line Integrals relate to conservative vector fields?
5. What are some physical interpretations of conservative vector fields?

**Tip:** In vector calculus, always check the conditions under which theorems are applied, such as ensuring the vector field is defined over a simply connected domain when checking for conservativeness.

Learn how to determine the value of k that makes the vector field F(x,y,z) = (12xy^2z + 3y)i + (12x^2yz + 3x)j + k2x conservative using vector calculus. Detailed step-by-step solution with partial derivatives and curl calculations.

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