Question

Knowing the components of the fields \(a\) and \(b\) in bome coordinate system, find the components of their commutator.

Short answer

Answer: The components of the commutator of vector fields a and b are (\(c_1\), \(c_2\), \(c_3\)) with: \(c_1\)= \((a_2 * b_3 - a_3 * b_2)\), \(c_2\)= \((a_3 * b_1 - a_1 * b_3)\), \(c_3\)= \((a_1 * b_2 - a_2 * b_1)\).

Step-by-step solution

Identify the components of the vector fields a and b

Let's assume the components of the vector fields a and b are given as follows: a = (\(a_1\), \(a_2\), \(a_3\)) and b = (\(b_1\), \(b_2\), \(b_3\)) where \(a_1\), \(a_2\), \(a_3\) and \(b_1\), \(b_2\), \(b_3\) are functions of the coordinates of some unspecified coordinate system.

Compute the cross product of a and b

The cross product (or commutator) of two vector fields a and b can be computed in terms of its components as follows: c = a × b = (\(c_1\), \(c_2\), \(c_3\)) where \(c_1\)= \((a_2 * b_3 - a_3 * b_2)\), \(c_2\)= \((a_3 * b_1 - a_1 * b_3)\), \(c_3\)= \((a_1 * b_2 - a_2 * b_1)\).

Find the components of the commutator

Now, plug in the components of the vector fields a and b from Step 1 to get the components of the commutator: \(c_1\)= \((a_2 * b_3 - a_3 * b_2)\) \(c_2\)= \((a_3 * b_1 - a_1 * b_3)\) \(c_3\)= \((a_1 * b_2 - a_2 * b_1)\). Now we have found the components of the commutator vector field c as (\(c_1\), \(c_2\), \(c_3\)).

Key concepts

Vector Field Components

To start, let's explore vector field components. Think of a vector field as a map that assigns a vector to every point in space. These vectors can represent anything from wind speeds in the atmosphere to magnetic forces around a magnet.

Each vector in the field can be broken down into components along the axes of a coordinate system. In a three-dimensional space, these components are typically labeled as \(a_1\), \(a_2\), and \(a_3\) for vector field \(a\), and correspondingly \(b_1\), \(b_2\), and \(b_3\) for vector field \(b\). The values of these components at any given point in space can vary, which allows the vector field to represent complex, multi-directional phenomena like fluid flow or electromagnetism.

By identifying and understanding these components, we can study and visualize the behavior of the vector field in great detail, making it one of the fundamental concepts in vector calculus and physics.

Cross Product

In the context of vector calculus, the cross product is a critical operation. It allows us to find a vector that is perpendicular to two other vectors. This new vector's magnitude is proportional to the area of the parallelogram spanned by the original vectors, providing a sense of their 'combined spread' in space.

The cross product is crucial while dealing with physical concepts like torque and angular momentum. It is defined using the components of vectors as shown in the exercise solution. Remember, the cross product is not commutative; that is, \(a \times b \) is not the same as \(b \times a\) - their directions are opposite, which aligns with the right-hand rule used in physics and mathematics.

Coordinate Systems

A coordinate system offers a reference framework to precisely define the position of points in space. Common systems include Cartesian (rectangular), cylindrical, and spherical coordinates. Each system is best suited for problems with certain symmetries and properties.

In a Cartesian coordinate system, the vector field components \(a_1\), \(a_2\), \(a_3\) and \(b_1\), \(b_2\), \(b_3\) correspond to the x, y, and z-axis respectively, which makes visualization and computation direct and straightforward. Understanding how vector fields behave under different coordinate systems is instrumental in fields like engineering and physics, where problems can be massively simplified by choosing the appropriate system.

Differential Equations

Vector fields are often described by differential equations, which are mathematical equations relating functions and their derivatives. Differential equations are integral to modeling the rate at which changes occur in physical systems, whether it's the growth of a population or the motion of a particle in a fluid.

When solving physics or engineering problems, we might need to find the field that satisfies certain differential equations, reflecting the underlying physical laws. For instance, Maxwell's equations for electromagnetic fields or Navier-Stokes equations for fluid dynamics. The ability to solve these equations links abstract mathematics with tangible physical phenomena.