Find the value of m if F = mx i - 5y j + 2z k is a Solenoidal vector.

Steps to Solve

1. Identify the vector field and its divergence

The given vector field is

$$ \mathbf{F} = mx \hat{i} - 5y \hat{j} + 2z \hat{k}. $$

We need to find the divergence of this vector field.

2. Calculate the divergence of the vector field

The divergence of a vector field ( \mathbf{F} = P \hat{i} + Q \hat{j} + R \hat{k} ) is given by

$$ \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}. $$

For our vector field:

• ( P = mx )
• ( Q = -5y )
• ( R = 2z )

Now calculate each partial derivative:

• ( \frac{\partial P}{\partial x} = \frac{\partial (mx)}{\partial x} = m )

• ( \frac{\partial Q}{\partial y} = \frac{\partial (-5y)}{\partial y} = -5 )

• ( \frac{\partial R}{\partial z} = \frac{\partial (2z)}{\partial z} = 2 )

3. Combine the results into the divergence formula

Now substitute these derivatives into the divergence formula:

$$ \nabla \cdot \mathbf{F} = m - 5 + 2 = m - 3. $$

4. Set the divergence equal to zero

Since the vector field is solenoidal, we set the divergence to zero:

$$ m - 3 = 0. $$

5. Solve for m

Now solve the equation for ( m ):

$$ m = 3. $$