Calculating the Curl of a Vector Field

The question asks us to find the curl of the given vector field \( \mathbf{r} = 2xy\mathbf{i} + \mathbf{j} + k\mathbf{l} \). In standard vector calculus, a vector field in 3D space is represented in terms of the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), corresponding to the x, y, and z directions, respectively. The term \( k\mathbf{l} \) seems non-standard. Given the options provided, which involve \( \mathbf{k} \), it is highly probable that the vector field is intended to be written as \( \mathbf{r} = 2xy\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} \), or perhaps with a constant \(k\) multiplying the \( \mathbf{k} \) vector, like \( \mathbf{r} = 2xy\mathbf{i} + 1\mathbf{j} + k\mathbf{k} \). However, for calculating the curl, partial derivatives with respect to x, y, and z are needed. If we assume \( \mathbf{r} = 2xy\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} \), the components are \( P = 2xy \), \( Q = 1 \), and \( R = 0 \). Let's proceed with this common representation based on the nature of the problem and options.

Definition of Curl

The curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is a vector operator that describes the infinitesimal rotation of the vector field in 3D. It is defined as:

Expanding this determinant gives:

Applying the Formula to r = 2xyi + j + 0k

We have \( P = 2xy \), \( Q = 1 \), and \( R = 0 \). Now we calculate the required partial derivatives:

• \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(2xy) = 2x \)
• \( \frac{\partial P}{\partial z} = \frac{\partial}{\partial z}(2xy) = 0 \)
• \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(1) = 0 \)
• \( \frac{\partial Q}{\partial z} = \frac{\partial}{\partial z}(1) = 0 \)
• \( \frac{\partial R}{\partial x} = \frac{\partial}{\partial x}(0) = 0 \)
• \( \frac{\partial R}{\partial y} = \frac{\partial}{\partial y}(0) = 0 \)

Calculating the Components of curl(r)

Now substitute these partial derivatives into the curl formula:

• \( \mathbf{i} \)-component: \( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 0 - 0 = 0 \)
• \( \mathbf{j} \)-component: \( -\left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right) = -(0 - 0) = 0 \)
• \( \mathbf{k} \)-component: \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 - 2x = -2x \)

Result of the Curl Calculation

Combining the components, the curl of \( \mathbf{r} \) is:

This result matches one of the given options. The calculation demonstrates the process of finding the curl using partial derivatives of the vector field components.

Thus, the curl of \( \mathbf{r} \) is \( -2x\mathbf{k} \).

Revision Table: Curl of Vector Fields

Additional Information: Vector Calculus Operations

Vector calculus involves several fundamental operations on scalar and vector fields, including gradient, divergence, and curl. These operators use the del operator \( \nabla \).

• Gradient (\( \nabla f \)): Operates on a scalar field \( f \) to produce a vector field that points in the direction of the greatest rate of increase of \( f \).

  \( \nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k} \)

• Divergence (\( \nabla \cdot \mathbf{F} \)): Operates on a vector field \( \mathbf{F} \) to produce a scalar field that measures the magnitude of a source or sink at a given point.

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

• Curl (\( \nabla \times \mathbf{F} \)): Operates on a vector field \( \mathbf{F} \) to produce a vector field that measures the tendency of the field to rotate about a point.

  \( \nabla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} - \left( \frac{\partial R}{\partial x} - \frac{\partial P}{\partial z} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k} \)

These operations are crucial in physics and engineering, particularly in fluid dynamics, electromagnetism, and field theory.