The magnitude of the curl of the vector F = 2xi + 3yj + 4zk, is

Understand the Problem

The question is asking for the magnitude of the curl of a given vector field defined by the function F = 2xi + 3yj + 4zk. The concept of curl is a measure of the rotation of a vector field, and finding its magnitude involves applying vector calculus.

Answer

The magnitude of the curl is $0$.

Steps to Solve

Identify the Vector Field We have the vector field ( \mathbf{F} = 2x \mathbf{i} + 3y \mathbf{j} + 4z \mathbf{k} ).
Recall the Curl Formula The curl of the vector field ( \mathbf{F} ) is given by: $$ \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ 2x & 3y & 4z \end{vmatrix} $$
Calculate the Determinant Expanding the determinant, we get:
- For ( \mathbf{i} ): $$ \frac{\partial}{\partial y}(4z) - \frac{\partial}{\partial z}(3y) = 0 - 0 = 0 $$
- For ( \mathbf{j} ): $$ \frac{\partial}{\partial z}(2x) - \frac{\partial}{\partial x}(4z) = 0 - 0 = 0 $$
- For ( \mathbf{k} ): $$ \frac{\partial}{\partial x}(3y) - \frac{\partial}{\partial y}(2x) = 0 - 0 = 0 $$
Thus, $$ \nabla \times \mathbf{F} = 0 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} = \mathbf{0} $$
Find the Magnitude of the Curl The magnitude of the curl is: $$ |\nabla \times \mathbf{F}| = |\mathbf{0}| = 0 $$

The magnitude of the curl of the vector field ( \mathbf{F} = 2x \mathbf{i} + 3y \mathbf{j} + 4z \mathbf{k} ) is ( 0 ).

More Information

The curl of a vector field provides insight into its rotational characteristics. In this case, since the vector field is linear without any dependence on the spatial coordinates' cross-derivatives, the result shows no rotation.

Tips

A common mistake is to overlook that the curl of a linear vector field can be zero.
Ensure proper calculation of the determinant to avoid errors in signs or coefficients.

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