Problem 3 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The \( \mathbf{k} \)-component of \( \operatorname{curl}(\mathbf{F}) \) is \( y e^x - x e^y \).

Step 1: Recall the Definition of Curl in Two Dimensions

For a two-dimensional vector field \( \mathbf{F} = P(x,y) \mathbf{i} + Q(x,y) \mathbf{j} \), the curl, \( abla \times \mathbf{F} \), is a scalar field (actually its \( \mathbf{k} \)-component) given by the formula: \[ abla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}. \] Our goal is to compute this quantity for the given vector field.

Step 2: Identify Components of the Vector Field

In the vector field \( \mathbf{F} = (x e^y) \mathbf{i} + (y e^x) \mathbf{j} \), we identify:- \( P(x,y) = x e^y \) - \( Q(x,y) = y e^x \).These functions will be used to find the partial derivatives needed for the curl.

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

Calculate the partial derivative of \( Q(x,y) = y e^x \) with respect to \( x \): \[ \frac{\partial Q}{\partial x} = y \cdot \frac{\partial}{\partial x} (e^x) = y e^x. \]

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

Calculate the partial derivative of \( P(x,y) = x e^y \) with respect to \( y \): \[ \frac{\partial P}{\partial y} = x \cdot \frac{\partial}{\partial y} (e^y) = x e^y. \]

Step 5: Compute the Curl

Using the formula for curl, calculate: \[ abla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = y e^x - x e^y. \] This expression represents the \( \mathbf{k} \)-component of the curl of the vector field \( \mathbf{F} \).

Key Concepts

Partial Derivatives

In the realm of calculus, partial derivatives are essential when dealing with functions of more than one variable. They measure how a multivariable function changes as only one of the variables is altered. By keeping other variables constant, we can focus on the rate of change concerning a particular variable. For example, when we have a function \( f(x, y) \), the partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \). This represents how \( f \) changes as \( x \) changes, with \( y \) held constant. Applying this concept in vector calculus is crucial, as it allows us to determine the effect of changing individual components. This forms the backbone of operations like the curl, where we calculate derivatives of vector field components separately.

Two-Dimensional Vector Fields

A two-dimensional vector field is a function that assigns a vector to every point in a plane. It can be visualized as a collection of arrows, each with a direction and magnitude, indicating how a quantity like fluid velocity or electromagnetic force varies across a region. Such a field is typically represented as \( \mathbf{F}(x, y) = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \), where \( P(x, y) \) and \( Q(x, y) \) are functions of \( x \) and \( y \) and \( \mathbf{i}, \mathbf{j} \) are unit vectors along the axes. Understanding the behavior of these fields helps us analyze physical phenomena, as each location in the plane reveals characteristics of the vector field.

Vector Field Components

In vector calculus, vector fields are composed of components that define how the vector field changes spatially. Each vector field can be broken down into its components along coordinate axes. For a vector field \( \mathbf{F} = P(x, y) \mathbf{i} + Q(x, y) \mathbf{j} \), \( P(x, y) \) and \( Q(x, y) \) represent its projections onto the \( \mathbf{i} \) and \( \mathbf{j} \) directions, respectively. These components tell us the influence in each axis' direction and are essential for calculating operations like divergence and curl. For curl specifically, the difference in the rate of change of these components, measured by partial derivatives, determines the rotational tendency of the field.

Scalar Field

A scalar field is a function that assigns a scalar value to every point in space. In the context of vector calculus and operations such as curl in two dimensions, the result is often a scalar field, highlighting the scalar nature of the outcome. The curl of a two-dimensional vector field is not a vector but a scalar function of \( x \) and \( y \), summarizing the rotation at each point in a plane.This scalar value, derived from the Curl formula \( abla \times \mathbf{F} = \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \), provides insight into how much and in which direction the field twists. By calculating this for various points, one can comprehend the overall rotational effect within the vector field.