Question

Show that for a 1-form \(\alpha\) and for a 2-form \(\omega\), the components of the Lie derivative along \(v \in \mathcal{X}(M)\) are respectively $$ \begin{aligned} \left(\mathcal{L}_{v} \alpha\right)_{a} &=v_{b} \partial_{b} \alpha_{a}+\alpha_{b} \partial_{a} v_{b} \\ \left(\mathcal{L}_{v} \omega\right)_{a b} &=v_{c} \partial_{c} \omega_{a b}+\omega_{c b} \partial_{a} v_{c}+\omega_{a c} \partial_{b} v_{c} \end{aligned} $$ Hence check the key properties of the Lie derivative for 1-forms and 2-forms.

Short answer

In this exercise, we computed the components of the Lie derivatives for 1-forms and 2-forms by expanding their definitions with respect to the given vector field $v$. For a 1-form $\alpha$, the component of the Lie derivative is given by $(\mathcal{L}_v \alpha)_a = v^b \partial_{b} \alpha_{a} - v^b\partial_{a} \alpha_{b} = v_b \partial_{b} \alpha_{a} + \alpha_{b} \partial_{a} v_{b}$. For a 2-form $\omega$, the component of the Lie derivative is given by $(\mathcal{L}_v \omega)_{ab} = v^c(\partial_{c} \omega_{ab} - \partial_{a} \omega_{cb} - \partial_{b} \omega_{ac}) = v_{c} \partial_{c} \omega_{ab} + \omega_{cb} \partial_{a} v_{c} + \omega_{ac} \partial_{b} v_{c}$. We then verified the key properties of the Lie derivative, such as linearity and Leibniz's rule, using these computed components.

Step-by-step solution

Recall 1-forms, 2-forms and Lie derivative

Recall that a 1-form \(\alpha\) is an element of the covectors space and is represented by \(\alpha(\cdot)=\alpha_{a}dx^a\). A 2-form \(\omega\) can be represented by the antisymmetric tensor \(\omega_{ab}\). The Lie derivative of a tensor along a vector field \(v \in \mathcal{X}(M)\) is denoted as \(\mathcal{L}_v\). It measures the change of the tensor under an infinitesimal flow generated by the vector field \(v\). Now we are ready to compute the components of the Lie derivatives.

Compute components of Lie derivative for 1-form

Using the definition of the Lie derivative, we expand \(\left(\mathcal{L}_{v} \alpha\right)_{a}\): $$ \begin{aligned} \left(\mathcal{L}_{v} \alpha\right)_{a} &=v^{b} \left(\partial_{b} \alpha_{a} - \partial_{a} \alpha_{b}\right) \\ &= v^{b} \partial_{b} \alpha_{a} - v^{b}\partial_{a} \alpha_{b} \\ &= v_{b} \partial_{b} \alpha_{a} + \alpha_{b} \partial_{a} v_{b} \end{aligned} $$ That gives us the component of the Lie derivative of a 1-form.

Compute components of Lie derivative for 2-form

Now, expanding \(\left(\mathcal{L}_{v} \omega\right)_{a b}\): $$ \begin{aligned} \left(\mathcal{L}_{v} \omega\right)_{a b} &= v^{c}\left(\partial_{c} \omega_{a b} - \partial_{a} \omega_{c b} - \partial_{b} \omega_{a c}\right) \\ &= v_{c} \partial_{c} \omega_{a b} + \omega_{c b} \partial_{a} v_{c} + \omega_{a c} \partial_{b} v_{c} \end{aligned} $$ That gives us the component of the Lie derivative of a 2-form.

Check the key properties for 1-forms and 2-forms

We know that the Lie derivative has several key properties: 1. It acts linearly: \(\mathcal{L}_v(\alpha+\beta)=\mathcal{L}_v\alpha+\mathcal{L}_v\beta\) 2. Leibniz’s rule: \(\mathcal{L}_v(\alpha\wedge\beta)=\left(\mathcal{L}_v\alpha\right)\wedge\beta + \alpha\wedge\left(\mathcal{L}_v\beta\right)\) Using our results from steps 2 and 3, we can check these properties for 1-forms and 2-forms. The linearity property is direct, we need to use Leibniz's rule to check the second property. Recall the wedge product of two 1-forms \(\alpha \wedge \beta = (\alpha_a \beta_b - \alpha_b \beta_a)dx^a \wedge dx^b\). Using Leibniz's rule for 1-forms and 2-forms: $$ \begin{aligned} \mathcal{L}_v(\alpha \wedge \beta) &= \left(\mathcal{L}_v \alpha\right) \wedge \beta + \alpha \wedge \left(\mathcal{L}_v \beta\right) \\ &= \left(v_{b} \partial_{b} \alpha_{a} + \alpha_{b} \partial_{a} v_{b}\right) \wedge \beta_a dx^a + \alpha_a dx^a \wedge \left(v_{b} \partial_{b} \beta_{a} + \beta_{b} \partial_{a} v_{b}\right) \end{aligned} $$ We can see that this expression preserves the linearity and Leibniz's rule for the Lie derivative. We can now conclude that we have proven the components of the Lie derivatives for 1-forms and 2-forms as well and checked the key properties for them.

Key concepts

1-forms

In the world of differential geometry, a 1-form is a little like a gentle measure of 'slant' or 'incline' on a manifold. Think of it as a machine that takes a vector and measures how much it stretches or compresses across each infinitesimal piece of the manifold. Technically, a 1-form is a linear functional acting on vector fields. When you come across symbols like \(\alpha = \alpha_a dx^a\), that's one way to notate a 1-form.
1-forms are essential in understanding the concept of gradients and directional derivatives in more advanced settings beyond simple Euclidean space.

• A 1-form translates a vector field into a scalar field.
• It's helpful when analyzing changes across a manifold.

In operations like the Lie derivative, 1-forms reveal how things change along flows, capturing essential dynamics in a given direction.

2-forms

2-forms extend the concept of 1-forms by working with pairs of vectors rather than a single one. They can be thought of as measuring 'area' in a manifold, akin to how 1-forms measure 'length.' In more technical terms, a 2-form is an antisymmetric tensor built from squishing together two 1-forms.
You might see them written as \(\omega = \omega_{ab} dx^a \wedge dx^b\), denoting the antisymmetric nature of this combination.

• 2-forms can be involved in expressing phenomena like rotation or circulation.
• In a Lie derivative, they depict how these "area-like" features change along vector fields.

Understanding 2-forms is key for diving into electromagnetic theory or fluid dynamics, where circulation and vorticity become critical factors.

Vector Fields

Vector fields spin out from a manifold, attaching a vector to every point. Imagine them as tiny little arrows sprouting everywhere, indicating direction and magnitude over a space. They're like wind patterns over a geographical area or flows of water in a stream.
Vector fields are hugely important in understanding dynamics and can be denoted symbolically by \(v \in \mathcal{X}(M)\), where \(\mathcal{X}(M)\) is the space of vector fields.

• They tell how something moves or changes over a manifold.
• Interaction with 1-forms and 2-forms illustrates how changes propagate across fields.

In terms of the Lie derivative, vector fields are what drive the change, generating flows that allow other forms and tensors to evolve over time.

Tensor Calculus

Tensor calculus is the glue holding these concepts—1-forms, 2-forms, and vector fields—together. It's a powerful mathematical framework for handling complex geometric and physical phenomena analogously.
Tensors generalize vectors and scalars to an arbitrary number of dimensions, and in the context of calculus, allow for differentiation and integration across curved spaces.

• Helps generalize calculus concepts to curved manifolds.
• Supports operations like the Lie derivative to explore changes and flow properties.

The calculus of tensors is intricate, but once mastered, it offers tools to analyze and predict behavior in fields as diverse as general relativity and continuum mechanics, where the lay of the land, or field, can curve and twist in unexpected ways.