Question

Show that when the group acts only on the single dependent variable $u$, the prolongation of $\mathbf{v}=U{\partial }_u$ is given by $${\mathrm{pr}}^{(\mathrm{I})}\mathbf{v}=\mathbf{v}+\sum _{j=1}^p{U}_j\frac{\partial }{\partial u_j},\text{ where }{U}_j=\frac{\partial U}{\partial x^j}+u_j\frac{\partial U}{\partial u}.$$

Short answer

The prolongation of the vector $\mathbf{v}=U{\partial }_u$ is given by $p{r}^{(I)}\mathbf{v}=U{\partial }_u+\sum _{j=1}^p(\frac{\partial U}{\partial x^j}+u_j\frac{\partial U}{\partial u})\frac{\partial }{\partial u_j}.$

Step-by-step solution

Define the Prolongation of the Vector

The prolongation of the vector $\mathbf{v}$ is defined as $p{r}^{(I)}\mathbf{v}$. This operation expands the vector $\mathbf{v}$ into a new space.

Substitute the Definition of the Vector $\mathbf{v}$

We know that $\mathbf{v}=U{\partial }_u$. So, by substituting this into $p{r}^{(I)}\mathbf{v}$, we get $p{r}^{(I)}\mathbf{v}=p{r}^{(I)}(U{\partial }_u).$

Expand the Prolongation using the given formula

The formula given in the task expands the prolongation as follows: $p{r}^{(I)}\mathbf{v}=\mathbf{v}+\sum _{j=1}^p{U}_j\frac{\partial }{\partial u_j}$. Substitute $\mathbf{v}=U{\partial }_u$ into this formula to get: $p{r}^{(I)}\mathbf{v}=U{\partial }_u+\sum _{j=1}^p{U}_j\frac{\partial }{\partial u_j}$.

Derive $U_j$

In the formula, $U_j$ is given by the derivative $U_j=\frac{\partial U}{\partial x^j}+u_j\frac{\partial U}{\partial u}$. Substitute this definition back into our formula from step 3 to obtain the final result.

Key concepts

Prolongation of Vector Fields

The concept of the prolongation of vector fields is critical to understanding the geometry of differential equations. To make it more approachable, let’s picture vector fields as arrows attached to points in space, indicating the direction and magnitude of a possible movement. Now when we say 'prolongation', think of it as an extension of these arrows into a higher-dimensional space, much like adding an extra dimension to a picture to give it more context.

In the exercise, we are working with a vector field which itself is defined in terms of the variable it's derived from, in this case, $U$ being a function that depends on $u$. The prolongation of this vector field involves taking into consideration how the function $U$ changes not only with $u$ but also with its derivatives, which are symbolized by $u_j$. Therefore, the 'prolonged' vector field includes additional components $U_j$, each representing the rate of change of $U$ with respect to different coordinates and the dependent variable $u$.

By considering the prolongation, we capture more information about the behavior of the system we are examining, allowing us to apply our analysis to situations that may involve higher derivatives, such as acceleration in physics or curvature in geometry.

Partial Derivatives

Partial derivatives are one of the foundational tools in the toolkit of calculus, particularly when dealing with multivariable functions. They measure how a function changes as you vary one of its inputs, holding the others constant, much like checking how steep a hill is by looking only directly north, ignoring the rest of the terrain.

This concept is key to our exercise, as the coefficients $U_j$ in the prolongation formula are calculated using partial derivatives. To find $U_j$, you compute how the function $U$ changes as you wiggle a little in the $x^j$ direction, then adjust this by how $U$ would change if our dependent variable $u$ were to change — this second part incorporates the influence of $u$ on the overall change in $U$ through its own partial derivative.

These partial derivatives encapsulate the local behavior of the function, giving us a zoomed-in view of the landscape that is our problem. By understanding how $U$ responds to changes in various directions, we become better equipped to predict and describe the behavior of the system described by our vector field $\mathbf{v}$.

Dependent Variable Transformation

Transforming dependent variables can be an intriguing concept, as it can significantly alter the way we view a problem. In a physical sense, think of it as changing your perspective: you're observing the action from a completely different angle, which can reveal new insights.

In the context of our exercise, the dependent variable $u$ is subject to transformation through the group action. This means that $u$ is not static; rather, it evolves according to certain rules defined by the transformation group. The prolongation of our vector field $\mathbf{v}$ thus takes into account these possible transformations of $u$ and adapts the vector field accordingly.

By including the derivatives $u_j$ as components in the prolonged vector field, the exercise exemplifies how transformations of the dependent variable invoke corresponding changes in the vector field, capturing the essence of how $u$ might behave under different scenarios. This understanding of dependent variable transformation is crucial for advancing in differential geometry and various applied mathematics fields.