Problem 4 - Thomas Calculus

Question

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=\left(x^{2} y\right) \mathbf{i}+\left(x y^{2}\right) \mathbf{j}\)

Step-by-Step Solution

Answer

The \(\mathbf{k}\)-component of the curl is \(y^2 - x^2\).

Step 1: Recall the Curl Formula in 2D

In two dimensions, the curl of a vector field \( \mathbf{F} = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \) is given by \( \operatorname{curl}(\mathbf{F}) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \). The result is a scalar, which corresponds to the \(\mathbf{k}\)-component.

Step 2: Identify P and Q in the Vector Field

The given vector field is \( \mathbf{F} = (x^2 y) \mathbf{i} + (xy^2) \mathbf{j} \). Here, \( P(x, y) = x^2 y \) and \( Q(x, y) = xy^2 \).

Step 3: Compute Partial Derivative of Q with Respect to x

The partial derivative of \( Q = xy^2 \) with respect to \( x \) is \( \frac{\partial Q}{\partial x} = y^2 \).

Step 4: Compute Partial Derivative of P with Respect to y

The partial derivative of \( P = x^2 y \) with respect to \( y \) is \( \frac{\partial P}{\partial y} = x^2 \).

Step 5: Calculate the k-component of the Curl

Insert the derivatives from Steps 3 and 4 into the curl formula: \( \operatorname{curl}(\mathbf{F}) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = y^2 - x^2 \). Thus, the \( \mathbf{k} \)-component of the curl is \( y^2 - x^2 \).

Key Concepts

Partial Derivatives

Partial derivatives are an essential concept in multivariable calculus. They measure how a function changes as one of its independent variables changes, while all other variables are held constant.
In the context of a vector field, partial derivatives help us understand the behavior of each component of the field in relation to the spatial coordinates.
Let's consider the vector field \( \mathbf{F} = (x^2y) \mathbf{i} + (xy^2) \mathbf{j} \). Here, we want to find how the function changes with respect to each variable.

• To find \( \frac{\partial Q}{\partial x} \) where \( Q = xy^2 \), differentiate \( Q \) while treating \( y \) as a constant. Hence \( \frac{\partial Q}{\partial x} = y^2 \).
• To find \( \frac{\partial P}{\partial y} \) where \( P = x^2y \), differentiate \( P \) with respect to \( y \) and treat \( x \) as a constant, resulting in \( \frac{\partial P}{\partial y} = x^2 \).

These derivatives are crucial when calculating the curl of a vector field.

2D Vector Field

A 2D vector field is a function that assigns a two-dimensional vector to every point in a plane. It's often represented as \( \mathbf{F} = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \), where \( P \) and \( Q \) are components of the field depending on \( x \) and \( y \).
These fields can be visualized as arrows on a plane, indicating both direction and magnitude at each point.
Understanding vector fields helps in analyzing phenomena like fluid flow, where each vector might represent speed and direction of flow at a point in the fluid.
In our example, the vector field is \( \mathbf{F} = (x^2y) \mathbf{i} + (xy^2) \mathbf{j} \). This shows how vectors are constructed from their components:

• \( (x^2y) \mathbf{i} \) indicates changes along the x-axis, influenced by both x and y.
• \( (xy^2) \mathbf{j} \) illustrates how changes occur along the y-axis, again depending on both variables.

Such fields are foundational to vector calculus and applications like electromagnetism and fluid dynamics.

k-component

In the context of 2D vector fields, the \( \mathbf{k} \)-component refers to the result obtained from the calculation of the curl. The curl measures the tendency of rotation at any point within the vector field.
The notation \( \mathbf{k} \) has its origins in three-dimensional vectors, where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) represent the unit vectors in the x, y, and z directions respectively.
In two dimensions, curl results in a scalar that behaves symbolically like the \( \mathbf{k} \)-component in 3D, even though we are working in 2D.
The formula for curl in 2D is \( \text{curl}(\mathbf{F}) = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \).
Using our earlier calculations:

• \( \frac{\partial Q}{\partial x} = y^2 \)
• \( \frac{\partial P}{\partial y} = x^2 \)

Thus, the \( \mathbf{k} \)-component of the curl is \( y^2 - x^2 \). This tells us how the field is rotating around a point, with positive values indicating counter-clockwise rotation and negative values indicating clockwise rotation.