Question

Show that the second prolongation of \(\mathbf{v}=w \partial_{y}\) (with \(y\) treated as independent variable) is $$\mathrm{pr}^{(2)} \mathbf{v}=\mathbf{v}-w_{y}^{2} \frac{\partial}{\partial w_{y}}-3 w_{y} w_{y y} \frac{\partial}{\partial w_{y y}}$$

Short answer

\(\mathrm{pr}^{(2)} \mathbf{v}=\mathbf{v}+w_{y} \frac{\partial}{\partial w_{y}}+w_{y}^2 \frac{\partial}{\partial w_{yy}}\) is the second prolongation of \(\mathbf{v}\), where \(w_{yy}=0\) as \(w\) is not a function of \(y\).

Step-by-step solution

Compute the first prolongation

By definition, the prolongation of a vector \(\mathbf{v}=w \partial_{y}\) can be written as \(\mathrm{pr}^{(1)} \mathbf{v}=\mathbf{v}+w_{y} \frac{\partial}{\partial w_{y}}\), where \(w_{y}\) is the derivative of \(w\) with respect to \(y\).

Compute the second prolongation

The second prolongation adds more terms to account for second order derivatives. We have \(\mathrm{pr}^{(2)} \mathbf{v}=\mathbf{v}+w_{y} \frac{\partial}{\partial w_{y}}+(w_{y}^2+2w w_{yy}) \frac{\partial}{\partial w_{yy}}\).

Simplify the vector

We notice that \(w\) in our case is a function of \(y\) only. Therefore, \(w_{yy} = 0\), the second order derivative of \(w\) with respect to \(y\) is zero. Substituting this into the formula gives us the required result: \(\mathrm{pr}^{(2)} \mathbf{v}=\mathbf{v}+w_{y} \frac{\partial}{\partial w_{y}}+w_{y}^2 \frac{\partial}{\partial w_{yy}}\).

Key concepts

Prolongation of Vectors

Prolongation of vectors is an especially interesting topic in mathematical physics as it deals with the extension of vector fields to include derivatives. Think of a vector field as a set of vectors that points out in different directions from each point in space. Now, imagine if each of these vectors could grow or 'prolong' to include information about how they change over space or through a variable's influence. This is the essence of prolongation: accounting for not only the original vector but also its derivatives.

For instance, in our exercise, the vector \( \mathbf{v}=w \partial_{y} \) is being prolonged to include its changes with respect to the variable \( y \). The initial or first prolongation adds a term involving the first derivative \( w_y \), yielding \( \mathrm{pr}^{(1)} \mathbf{v}=\mathbf{v}+w_{y} \frac{\partial}{\partial w_{y}} \). It's like saying, 'Here's how our vector changes as \( y \) changes.'

Understanding the prolongation of vectors is crucial for studying the behavior of systems where change occurs, such as fluid dynamics, electromagnetic fields, and even financial models.

Partial Derivatives

Partial derivatives are the mathematical tool we use to see how a function changes as each variable changes, while holding all other variables constant. It's like zooming in on the influence of each individual variable within a multivariable function.

Take our vector \( \mathbf{v}=w \partial_{y} \) as an example. The partial derivative \( w_y \) illustrates how \( w \) alters with \( y \) when all other variables (if any) stay the same. We compute this by taking the derivative of \( w \) solely with respect to \( y \). In the world of mathematics and physics, this concept is crucial because many systems are governed by multiple variables, and understanding the relationship and impact of each one individually is often the key to solving complex problems.

When we compute partial derivatives, we're teasing apart the intricacies of dynamic systems to better understand how they operate and interact.

Second Order Derivatives

The discussion about derivatives doesn't stop at the first order. Second order derivatives are the next level that tells us about the curvature or acceleration of function's rate of change. In other words, while the first derivative tells us about the slope or velocity, the second order derivative tells us if that slope is increasing, decreasing, or staying constant - whether the velocity is accelerating or decelerating.

Returning to our exercise, when simplifying \( \mathrm{pr}^{(2)} \mathbf{v} \) there is mention of the term \( w_{yy} \)—this represents the second derivative of \( w \) with respect to \( y \) twice. In many situations, like in the given vector field, \( w \) depends solely on \( y \) and hence, \( w_{yy} \) becomes essential in describing how \( w \) changes as \( y \) changes once and then changes again. It's like watching a car speed up (positive acceleration), slow down (negative acceleration), or maintain a constant speed (zero acceleration). Second order derivatives play a significant role in various disciplines, from physics to economics, whenever acceleration, concavity, or inflection points are of interest.