Calculate the value of div F for F = x²yi (z³-3x) + 4y²k

The given vector field \( \mathbf{F} \) is:

\[
\mathbf{F} = x^2y \hat{i} + (z^3 - 3x) \hat{j} + 4y^2 \hat{k}
\]

We are asked to compute the divergence of \( \mathbf{F} \), denoted by \( \text{div} \, \mathbf{F} \), which is defined as the dot product of the del operator \( \nabla \) with the vector field \( \mathbf{F} \). In three dimensions, the divergence is given by:

\[
\text{div} \, \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}
\]

Here, \( F_1 = x^2y \), \( F_2 = z^3 - 3x \), and \( F_3 = 4y^2 \).

Now, we calculate each partial derivative:

1. \( \frac{\partial F_1}{\partial x} = \frac{\partial}{\partial x}(x^2y) = 2xy \)
2. \( \frac{\partial F_2}{\partial y} = \frac{\partial}{\partial y}(z^3 - 3x) = 0 \) (since there is no \( y \)-dependence in \( F_2 \))
3. \( \frac{\partial F_3}{\partial z} = \frac{\partial}{\partial z}(4y^2) = 0 \) (since there is no \( z \)-dependence in \( F_3 \))

Thus, the divergence of \( \mathbf{F} \) is:

\[
\text{div} \, \mathbf{F} = 2xy + 0 + 0 = 2xy
\]

So, the value of \( \text{div} \, \mathbf{F} \) is \( 2xy \).

Would you like further details or clarifications?

Here are five related questions for further practice:

1. How would you compute the curl of the vector field \( \mathbf{F} \)?
2. What is the divergence of \( \mathbf{F} \) if \( \mathbf{F} = xz\hat{i} + y^2\hat{j} + 3z\hat{k} \)?
3. How would the divergence change if \( \mathbf{F} \) contained exponential terms?
4. What is the physical significance of the divergence of a vector field in fluid dynamics?
5. How would you calculate the divergence of \( \mathbf{F} \) in cylindrical coordinates?

**Tip:** The divergence of a vector field gives insight into whether there is a net "source" or "sink" at any point in the field.

Learn how to compute the divergence of the vector field F = x²y i + (z³-3x) j + 4y² k. This step-by-step guide explores key vector calculus concepts like divergence and vector fields, focusing on applying the formula div F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z. Suitable for advanced students studying Calculus III or Multivariable Calculus.

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