Problem 244 - Calculus Volume 3

Question

Use a computer algebra system to find the curl of the given vector fields. $$ \mathbf{F}(x, y, z)=\sin (x-y) \mathbf{i}+\sin (y-z) \mathbf{j}+\sin (z-x) \mathbf{k} $$

Step-by-Step Solution

Answer

The curl of \( \mathbf{F} \) is \( \nabla \times \mathbf{F} = \cos(y-z)\mathbf{i} + \cos(z-x)\mathbf{j} + \cos(x-y)\mathbf{k} \).

Step 1: Introduction to Curl

The curl of a vector field \( \mathbf{F} = M\mathbf{i} + N\mathbf{j} + P\mathbf{k} \) is determined using the formula \( abla \times \mathbf{F} = \left( \frac{\partial P}{\partial y} - \frac{\partial N}{\partial z} \right)\mathbf{i} + \left( \frac{\partial M}{\partial z} - \frac{\partial P}{\partial x} \right)\mathbf{j} + \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right)\mathbf{k} \). We will find the curl of the given vector field using this formula.

Step 2: Identify Component Functions

The vector field given is \( \mathbf{F}(x, y, z)=\sin (x-y) \mathbf{i}+\sin (y-z) \mathbf{j}+\sin (z-x)\mathbf{k} \). Here, \( M(x,y,z)=\sin(x-y) \), \( N(x,y,z)=\sin(y-z) \), and \( P(x,y,z)=\sin(z-x) \).

Step 3: Calculate Partial Derivatives

Calculate the necessary partial derivatives: - \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}[\sin(z-x)] = 0 \) - \( \frac{\partial N}{\partial z} = \frac{\partial}{\partial z}[\sin(y-z)] = -\cos(y-z) \) - \( \frac{\partial M}{\partial z} = \frac{\partial}{\partial z}[\sin(x-y)] = 0 \) - \( \frac{\partial P}{\partial x} = \frac{\partial}{\partial x}[\sin(z-x)] = -\cos(z-x) \) - \( \frac{\partial N}{\partial x} = \frac{\partial}{\partial x}[\sin(y-z)] = 0 \) - \( \frac{\partial M}{\partial y} = \frac{\partial}{\partial y}[\sin(x-y)] = -\cos(x-y) \)

Step 4: Substitute Partial Derivatives into Curl Formula

Now, substitute the values into the curl formula: - \( A = \frac{\partial P}{\partial y} - \frac{\partial N}{\partial z} = 0 - (-\cos(y-z)) = \cos(y-z) \) - \( B = \frac{\partial M}{\partial z} - \frac{\partial P}{\partial x} = 0 - (-\cos(z-x)) = \cos(z-x) \) - \( C = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 0 - (-\cos(x-y)) = \cos(x-y) \)

Step 5: Write the Curl

Thus, the curl of the vector field \( abla \times \mathbf{F} \) is: \[ abla \times \mathbf{F} = \cos(y-z)\mathbf{i} + \cos(z-x)\mathbf{j} + \cos(x-y)\mathbf{k} \]

Key Concepts

Vector Fields

The concept of vector fields is fundamental in vector calculus. A vector field assigns a vector to every point in space. It's like giving each point a direction and magnitude. For instance, wind flow across an area can be represented by a vector field, where each arrow shows the direction and strength of the wind at that point.
A vector field \( \mathbf{F}(x, y, z) \)in three dimensions is defined by its components:

• \( M(x, y, z) \)giving the \( x \)-component,
• \( N(x, y, z) \)giving the \( y \)-component,
• \( P(x, y, z) \)giving the \( z \)-component.

In the given problem, the vector field is \( \mathbf{F}(x, y, z) = \sin(x-y)\mathbf{i} + \sin(y-z)\mathbf{j} + \sin(z-x)\mathbf{k} \).This means every point \( (x, y, z) \)in space has a vector with components dependent on sines of the differences between the coordinates.

Curl

Curl is a concept that helps us understand the rotation effect within a vector field in three dimensions. Imagine the flow of water in a vortex, the curl measures how strongly and in which direction the vortex is rotating.
Mathematically, the curl of a vector field \( \mathbf{F} = M\mathbf{i} + N\mathbf{j} + P\mathbf{k} \)is given by the formula:

• \( abla \times \mathbf{F} = \left( \frac{\partial P}{\partial y} - \frac{\partial N}{\partial z} \right)\mathbf{i} + \left( \frac{\partial M}{\partial z} - \frac{\partial P}{\partial x} \right)\mathbf{j} + \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right)\mathbf{k} \)

This formula calculates the curl by finding specific partial derivatives and then combining them. In simple terms, it checks how each component of the vector field rotates around the axes. For the given exercise, the computed result was \( \cos(y-z)\mathbf{i} + \cos(z-x)\mathbf{j} + \cos(x-y)\mathbf{k} \).This result indicates the amount of rotation at each point in the field.

Partial Derivatives

Understanding partial derivatives is crucial for working with vector fields and calculating curl. A partial derivative measures how a function changes as one of its variables changes, while all other variables are held constant.
For example, if you have a function \( f(x, y, z) = \sin(x - y) \),the partial derivative with respect to \( x \)would be computed as if \( y \)and \( z \)are constants. In simpler terms, you're looking at the slope of the function along the \( x \)direction only.
In the given problem, calculating each partial derivative step-by-step forms the backbone of finding the curl. For instance:

• \( \frac{\partial M}{\partial y} = -\cos(x-y) \)
• \( \frac{\partial N}{\partial z} = -\cos(y-z) \)

Such calculations tell you how each component of the field is changing in relation to each other.

Computer Algebra System

A computer algebra system (CAS) is a software tool that helps perform symbolic mathematics, including operations like differentiation, integration, and finding the curl of vector fields. These systems save a lot of time, especially for complex problems.
With a CAS, you simply enter your vector field equation and command it to compute the curl, which automates all the tedious manual calculations of partial derivatives and other operations.
Popular CAS options include:

• Mathematica
• Maple
• Wolfram Alpha

These tools are incredibly beneficial for students and professionals alike, providing accurate results and allowing users to focus on understanding concepts rather than just computation. In the context of the exercise, using a CAS helps efficiently compute the curl of the vector field, confirming results obtained through manual calculations.