. Show that the exterior square of a 1-form, or, in general, of a form of odd order, is equal to zero: \(\omega^{k} \wedge \omega^{k}=0\) if \(k\) is odd.

Differential forms are a powerful tool used in multivariable calculus, differential geometry, and physics. They generalize the concept of functions and can be integrated over paths, surfaces, and higher-dimensional objects. In essence, a 1-form can be visualized as associating a linear function to each point of a space, which takes in a vector and outputs a scalar.

For example, if you consider a smooth path through space, integrating a 1-form along this path gives you the concept of work done in physics. Higher-degree differential forms extend these ideas to surfaces and other manifolds, enabling the generalization of concepts like flux through a surface. The beauty of differential forms lies in their ability to capture geometrical and analytical information in a unified framework.

The exterior square of a form is essentially a construction that involves taking the wedge product of a differential form with itself. This operation is part of what mathematicians call the 'exterior algebra.' The exterior square is particularly interesting because it can provide insights into the properties of forms and the spaces they inhabit.

Consider a 1-form which might represent, in some sense, a directional gradient. When taking its exterior square, you're trying to combine that gradient with itself in a way governed by the rules of differential forms. As your intuition might suggest, and as the exercise confirms, doing this with odd-degree forms results in zero, which tells us that there's a fundamental property of 'non-repeatability' or antisymmetry at play within the algebraic structure of differential forms.

The anti-commutativity property is a critical aspect of the wedge product operation. It states that when you swap the order of two differential forms in a wedge product, the result is the same but with an opposite sign — if both the forms have an odd degree. This is not just a quirky mathematical rule but has practical implications. For instance, in physics, this property reflects the impossibility of certain combined states.

This anti-commutative nature prevents us from 'doubling' the effect of a field upon itself as it would suggest negating the original configuration. When carefully considered, this property becomes a cornerstone in understanding many concepts in modern geometry and physics, such as explaining why certain physical phenomena are orientation-dependent.

Mathematical proofs are the bedrock of mathematics. They are rigorous ways of demonstrating that certain statements are always true. The technique used in this exercise is known as a 'direct proof,' which involves applying known properties, like the anti-commutativity of the wedge product, to derive a straightforward conclusion.

In this case, the proof hinges on simple algebraic manipulation and understanding the properties of differential forms to arrive at a sound conclusion. This direct method is particularly elegant, combining logical reasoning with the structured nature of mathematical operations. Direct proof is just one tool in a vast toolbox that mathematicians use to shed light on the abstract relationships they study.