Question

Show that the exterior derivative of \(\omega=A d y d z+\) \(B d z d x+C d x d y\) is the three-form $$ \left(\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}\right) d x d y d z $$

Short answer

Question: Compute the exterior derivative of the 2-form \(\omega = A dy dz + B dz dx + C dx dy\) and show that it is equal to the given 3-form, involving partial derivatives of \(A\), \(B\), and \(C\). Answer: The exterior derivative of \(\omega\) is given by the following 3-form: $$ d\omega = \left( \frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial C}{\partial z} \right) dx dy dz $$

Step-by-step solution

Exterior Derivative Definition

The exterior derivative of a given differential form is the result of applying the exterior differentiation to it and denoted with "d". For a 2-form \(\omega\), its exterior derivative is a 3-form given by the sum of the derivatives of its coefficients with respect to the corresponding variables, multiplied by corresponding exterior products of basis 1-forms.

Compute the Exterior Derivatives of Coefficients

First, we compute the exterior derivatives of A, B, and C with respect to the variables x, y, and z: $$ \frac{\partial A}{\partial x} dx, \frac{\partial B}{\partial y} dy, \frac{\partial C}{\partial z} dz $$

Apply Exterior Differentiation to \(\omega\)

Now, let's compute the exterior derivative \(d\omega\) by applying the exterior difference rule to \(\omega = A dy dz + B dz dx + C dx dy\): $$ d\omega = d(A dy dz + B dz dx + C dx dy) $$

Distribute the Exterior Derivative

Distribute d to each term in \(\omega\): $$ d\omega = d(A dy dz) + d(B dz dx) + d(C dx dy) $$

Compute Exterior Derivatives of Individual Terms

Let's compute the exterior derivatives of each term: $$ d(A dy dz) = \frac{\partial A}{\partial x} dx dy dz \\ d(B dz dx) = \frac{\partial B}{\partial y} dy dz dx \\ d(C dx dy) = \frac{\partial C}{\partial z} dz dx dy $$

Rearrange the Basis 1-forms for the Second and Third Terms

We must rearrange the basis 1-forms for the second and third terms to make them in the same order \(dx dy dz\). Using the antisymmetry property (\(\alpha \wedge \beta = - \beta \wedge \alpha\)) of exterior product, we have: $$ dy dz dx = (-1)^2 dz dx dy = -dz dx dy \\ dz dx dy = (-1)^{\phantom{2}} dx dy dz = - dx dy dz $$

Combine the Terms to Obtain the Result

Now we combine the terms to obtain the 3-form: $$ d\omega = \frac{\partial A}{\partial x} dx dy dz - \frac{\partial B}{\partial y} dz dx dy - \frac{\partial C}{\partial z} dx dy dz \\ d\omega = \left( \frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial C}{\partial z} \right) dx dy dz $$ Therefore, we have shown that the exterior derivative of \(\omega = A dy dz + B dz dx + C dx dy\) is the three-form: $$ d\omega = \left( \frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial C}{\partial z} \right) dx dy dz $$

Key concepts

Differential Forms

Imagine you are walking through a field and you want to describe how the elevation changes under your feet with each step. You could use numbers to express these changes: up, down, steeper, or flatter. In mathematics, the tool we use for such descriptions can be called a 'differential form'.

A differential form is an algebraic construct used in calculus to generalize the concept of functions, integrating and differentials to higher dimensions. Simply put, it's a mathematical way to encode how a quantity changes as you move in different directions through space.

For example, \( dx \) and \( dy \) are basic 1-forms that represent infinitesimal changes in the x and y directions. You can think of them like tiny steps in these directions. When you see something like \( A dx \), it combines a function \( A \) with the 1-form \( dx \) and tells you how \( A \) changes as you move in the x-direction. Now, if you want to describe how something changes in two directions simultaneously, you use a 2-form, such as \( A dy \wedge dz \). This represents a tiny parallelogram expanding in the y and z directions. Higher-dimensional differential forms generalize this concept even further.

Antisymmetry Property

To understand the antisymmetry property, let's take a real-world example. Imagine having a piece of paper and marking two points that define a direction and distance. If you swap the points, the direction flips. This 'flipping' characteristic is an everyday manifestation of the antisymmetry property.

In the realm of differential forms, this property is crucial. When we talk about the exterior product (also denoted as \( \wedge \) ), which combines different 1-forms, its antisymmetry comes into play. Say, for example, \( dz \wedge dx \) is a differential form combining changes in z and x directions. Now, if we try to swap \( dz \), and \( dx \) to get \( dx \wedge dz \) the result is not the same; instead, it’s the negative of the original: \( -dz \wedge dx \).

This property implies that if you combine the same 1-form with itself, like \( dx \wedge dx \), the result is zero because the form can’t represent an area if both directions are the same. So, the antisymmetry property ensures that our differential forms make geometrical and physical sense.

Exterior Differentiation

Let's think of a recipe to add some 'spice' to a dish. In mathematics, the 'spice' we add to differential forms to extract new information is called exterior differentiation.

When you perform exterior differentiation, you're essentially taking a derivative of a differential form to produce a new form one degree higher. You're expanding the dimensionality of the information you have. For a 2-form \( \omega \) the exterior derivative \( d\omega \) is a 3-form, and it is found by taking the partial derivatives of the functions in front of your original 2-forms with respect to all possible variables, and combining them with the exterior product of the corresponding differentials.

The process, while sounding complex, adheres to rules similar to those of regular differentiation, such as the product rule. The key takeaway is that through exterior differentiation, we gain a wider picture of our mathematical 'landscape'- much like how adding just the right amount of spice reveals new flavors in a dish.